[{"data":1,"prerenderedAt":13},["ShallowReactive",2],{"$fZMeidOb0G9yiRDfJPEAtGvrZwNVG6N_lC4sfzWwnr3o":3,"$fcro8Rsr-VGomFGUjY0Vs0DR8jJ9dNBx9AfTEL-iVwZI":6},{"long":4,"short":5},"2c8e07faa515d74fe4caedd8f28fa7a408736f97","2c8e07f",{"body":7,"slug":8,"name":9,"page-name-search":10,"page-title":9,"categories":11},"\u003Csection class=\"unnumbered docname\">\n\u003Ch1 class=\"unnumbered\">Équations différentielles\u003C/h1>\n\u003C/section>\n\u003Cdiv class=\"doctitle\">\n\u003Cp>Équations différentielles\u003C/p>\n\u003C/div>\n\u003Cdiv class=\"doccategories\">\n\u003Cp>algebra\u003C/p>\n\u003C/div>\n\u003Cdiv class=\"docsummary\">\n\n\u003C/div>\n\u003Cp>\u003Cstrong>Restitution des connaissances\u003C/strong>\u003C/p>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 1\u003C/strong>. \u003Cem>Compléter les phrases\nci-dessous.\u003C/em>\u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Les solutions sur \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6889em;\">\u003C/span>\u003Cspan class=\"mord mathbb\">R\u003C/span>\u003C/span>\u003C/span>\u003C/span> de\nl’équation différentielle \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">−\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">a\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span> sont\nles fonctions du type \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.313em;\">\u003C/span>\u003Cspan class=\"minner\">⋯\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Une équation différentielle linéaire homogène du second ordre\nà coefficients constants est une équation du type \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.313em;\">\u003C/span>\u003Cspan class=\"minner\">⋯\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Les solutions sur \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6889em;\">\u003C/span>\u003Cspan class=\"mord mathbb\">R\u003C/span>\u003C/span>\u003C/span>\u003C/span> de\nl’équation différentielle \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0085em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ω\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span> sont les fonctions de la forme \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.313em;\">\u003C/span>\u003Cspan class=\"minner\">⋯\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>L’équation caractéristique de l’équation différentielle \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">a\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">b\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">cy\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span> est \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.313em;\">\u003C/span>\u003Cspan class=\"minner\">⋯\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 2\u003C/strong>. \u003Cem>Répondre par vrai ou faux.\u003C/em>\u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>L’équation différentielle \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003C/span>\u003C/span>\u003C/span>\na pour solution générale :\u003Cbr>\n\u003Cstrong>a)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">K\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6644em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\n\u003Cstrong>b)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7713em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\n\u003Cstrong>c)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7713em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">K\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>L’équation différentielle \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003C/span>\u003C/span>\u003C/span> a pour solution générale :\u003Cbr>\n\u003Cstrong>a)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7667em;vertical-align:-0.0833em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">A\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6644em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7713em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span> \u003Cstrong>b)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">A\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mop\">cos\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mop\">sin\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span> \u003Cstrong>c)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6644em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">A\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mop\">cos\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mop\">sin\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>La fonction \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.522em;vertical-align:-0.011em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">↦\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8141em;\">\u003C/span>\u003Cspan class=\"mord\">5\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mtight\">2\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\nest solution de l’équation différentielle :\u003Cbr>\n\u003Cstrong>a)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003C/span>\u003C/span>\u003C/span>\n\u003Cstrong>b)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord\">5\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003C/span>\u003C/span>\u003C/span>\n\u003Cstrong>c)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>La fonction \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.522em;vertical-align:-0.011em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">↦\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\">\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">−\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8141em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">4\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\nest solution de l’équation différentielle :\u003Cbr>\n\u003Cstrong>a)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">4\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">8\u003C/span>\u003C/span>\u003C/span>\u003C/span>\n\u003Cstrong>b)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">−\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">4\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">4\u003C/span>\u003C/span>\u003C/span>\u003C/span>\n\u003Cstrong>a)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">−\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">4\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">8\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>L’équation différentielle \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C/span>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7463em;\">\u003Cspan style=\"top:-2.786em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C/span>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mtight\">2\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose nulldelimiter\">\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span> a pour solution générale :\u003Cbr>\n\u003Cstrong>a)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\">\u003C/span>\u003Cspan class=\"mop\">ln\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C\u003C/span>\u003C/span>\u003C/span>\u003C/span> \u003Cstrong>b)\u003C/strong>\n\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C/span>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose nulldelimiter\">\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span> \u003Cstrong>c)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">K\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mop\">ln\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\n\u003C/div>\n\u003Cp>\u003Cstrong>Applications de règles ou de méthodes\u003C/strong>\u003C/p>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 3\u003C/strong>. \u003Cem>Résoudre les équations\ndifférentielles suivantes.\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>\u003Cstrong>a)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">4\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003C/span>\u003C/span>\u003C/span>\n\u003Cstrong>b)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C/span>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">3\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose nulldelimiter\">\u003C/span>\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>\u003Cstrong>c)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>\n\u003Cstrong>d)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">3\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">−\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 4\u003C/strong>. \u003Cem>Résoudre les équations\ndifférentielles suivantes.\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>\u003Cstrong>a)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">6\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">9\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span> \u003Cstrong>b)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">−\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">4\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">−\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">5\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>\u003Cstrong>c)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">3\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">10\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">3\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span> \u003Cstrong>d)\u003C/strong> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">7\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 5\u003C/strong>. \u003Cem>Résoudre sur \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6889em;\">\u003C/span>\u003Cspan class=\"mord mathbb\">R\u003C/span>\u003C/span>\u003C/span>\u003C/span> les équations différentielles\n:\u003Cstrong>a)\u003C/strong>   \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">4\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span> \u003Cstrong>b)\u003C/strong>  \n\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mord\">"\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">4\u003C/span>\u003C/span>\u003C/span>\u003C/span> \u003Cstrong>c)\u003C/strong>   \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mord\">"\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 6\u003C/strong>. \u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Résoudre dans \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6889em;\">\u003C/span>\u003Cspan class=\"mord mathbb\">R\u003C/span>\u003C/span>\u003C/span>\u003C/span> l’équation\ndifférentielle (E) : \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">3\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Déterminer la solution g de (E) dont la courbe représentative\nadmet pour tangente au point d’abscisse \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>, la droite\nd’équation \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">3\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 7\u003C/strong>. \u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Résoudre dans \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6889em;\">\u003C/span>\u003Cspan class=\"mord mathbb\">R\u003C/span>\u003C/span>\u003C/span>\u003C/span> l’équation\ndifférentielle (E) : \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C/span>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose nulldelimiter\">\u003C/span>\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C/span>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">3\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose nulldelimiter\">\u003C/span>\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">−\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Déterminer la solution \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g\u003C/span>\u003C/span>\u003C/span>\u003C/span> de (E) dont la\ncourbe représentative passe par le point A\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003Cspan class=\"mpunct\">,\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord\">1\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span> et\ndont la tangente en ce point est parallèle à l’axe des\nabscisses.\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 8\u003C/strong>. \u003Cem>On considère l’équation\ndifférentielle (E) suivante : \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">3\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>Déterminer les réels \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">a\u003C/span>\u003C/span>\u003C/span>\u003C/span> et \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">b\u003C/span>\u003C/span>\u003C/span>\u003C/span> pour que la fonction \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">h\u003C/span>\u003C/span>\u003C/span>\u003C/span> : \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.522em;vertical-align:-0.011em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">⟼\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">a\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">b\u003C/span>\u003C/span>\u003C/span>\u003C/span>, soit solution de (E).\u003C/em>\u003C/p>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 9\u003C/strong>. \u003Cem>Soit la fonction \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003C/span>\u003C/span>\u003C/span> définie sur \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6889em;\">\u003C/span>\u003Cspan class=\"mord mathbb\">R\u003C/span>\u003C/span>\u003C/span>\u003C/span> par : \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.522em;vertical-align:-0.011em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">⟼\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord\">1\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mtight\">2\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Déterminer les réels \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">a\u003C/span>\u003C/span>\u003C/span>\u003C/span> et \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">b\u003C/span>\u003C/span>\u003C/span>\u003C/span> pour que \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003C/span>\u003C/span>\u003C/span> soit solution de\nl’équation \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mord\">"\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">a\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">b\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>En déduire les primitives de \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003C/span>\u003C/span>\u003C/span> sur \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6889em;\">\u003C/span>\u003Cspan class=\"mord mathbb\">R\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 10\u003C/strong>. \u003Cem>Soient les équations\ndifférentielles : (E\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan>\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span> : \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span> et (E): \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7713em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mop\">cos\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Trouver les réels \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">a\u003C/span>\u003C/span>\u003C/span>\u003C/span> et \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">b\u003C/span>\u003C/span>\u003C/span>\u003C/span> pour que la fonction \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">h\u003C/span>\u003C/span>\u003C/span>\u003C/span> : \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.522em;vertical-align:-0.011em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">⟼\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0213em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">a\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mop\">cos\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">b\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mop\">sin\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">)\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\nsoit solution de (E).\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Démontrer qu’une fonction \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003C/span>\u003C/span>\u003C/span> est solution\nde (E) si et seulement si \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">−\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">h\u003C/span>\u003C/span>\u003C/span>\u003C/span> est solution de (E\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan>\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Résoudre (E\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan>\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Déduire des questions précédentes la solution générale de\n(E).\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Déterminer la solution \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g\u003C/span>\u003C/span>\u003C/span>\u003C/span> de (E) telle que\n\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">1\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 11\u003C/strong>. \u003Cem>On considère l’équation\ndifférentielle (E): \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8547em;vertical-align:-0.0833em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Monter que la fonction \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">h\u003C/span>\u003C/span>\u003C/span>\u003C/span> définie par :\n\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">h\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span> est une solution\nparticulière de (E).\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Résoudre dans \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6889em;\">\u003C/span>\u003Cspan class=\"mord mathbb\">R\u003C/span>\u003C/span>\u003C/span>\u003C/span> l’équation\ndifférentielle \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0019em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span> : \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Démontrer que \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003C/span>\u003C/span>\u003C/span> est solution de (E) si et\nseulement si \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">−\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">h\u003C/span>\u003C/span>\u003C/span>\u003C/span> est solution de \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0019em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>En déduire la solution générale \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003C/span>\u003C/span>\u003C/span> de\n(E).\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Déterminer la solution \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g\u003C/span>\u003C/span>\u003C/span>\u003C/span> de (E)\nsatisfaisant aux conditions initiales : \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">1\u003C/span>\u003C/span>\u003C/span>\u003C/span> et\n\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0019em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 12\u003C/strong>. \u003Cem>On considère sur \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6889em;\">\u003C/span>\u003Cspan class=\"mord mathbb\">R\u003C/span>\u003C/span>\u003C/span>\u003C/span> l’équation différentielle : (E): \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8141em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mtight\">2\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Vérifier que la fonction \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">:\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.522em;vertical-align:-0.011em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">⟼\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord\">1\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">e\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mtight\">2\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span> est une solution de(E).\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Démontrer qu’une \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g\u003C/span>\u003C/span>\u003C/span>\u003C/span> est solution de (E)\nsi et seulement si \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003C/span>\u003C/span>\u003C/span> est solution de \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Déduire des questions précédentes la solution générale de\n(E).\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 13\u003C/strong>. \u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Résoudre l’équation différentielle (E): \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">5\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Déterminer la solution de \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003C/span>\u003C/span>\u003C/span> qui vérifie\n\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">1\u003C/span>\u003C/span>\u003C/span>\u003C/span> et \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0019em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord\">1\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>On pose \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">F\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C/span>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">5\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose nulldelimiter\">\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">(\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Démontrer que \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">F\u003C/span>\u003C/span>\u003C/span>\u003C/span> est une primitive de \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003C/span>\u003C/span>\u003C/span> sur \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6889em;\">\u003C/span>\u003Cspan class=\"mord mathbb\">R\u003C/span>\u003C/span>\u003C/span>\u003C/span> puis expliciter \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">F\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>En déduire le calcul de \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.3978em;vertical-align:-0.3558em;\">\u003C/span>\u003Cspan class=\"mop\">\u003Cspan class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0006em;\">∫\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.042em;\">\u003Cspan style=\"top:-2.6442em;margin-left:-0.1945em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.5579em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen nulldelimiter sizing reset-size3 size6\">\u003C/span>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6915em;\">\u003Cspan style=\"top:-2.656em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.2255em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\">\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.384em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">π\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.344em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose nulldelimiter sizing reset-size3 size6\">\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3558em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">d\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\u003C/li>\n\u003C/ol>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 14\u003C/strong>. \u003Cem>Une note de musique est émise en\npinçant la corde d’une guitare électrique.\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>La puissance du son émis, initialement de 100 watts, diminue avec\nle temps t, mesuré en secondes.\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>On modélise par f(t) la puissance du son émis, exprimée en watt,\nt secondes après le pincement de la corde. Le son s’affaiblit à une\nvitesse proportionnelle à sa puissance, il a été établi que le\ncoefficient de proportionnalité est de \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">−\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003Cspan class=\"mpunct\">,\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord\">12\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Écrire l’équation différentielle traduisant la diminution de\nson.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Déterminer la fonction f solution de l’équation\ndifférentielle (E) qui vérifie la condition initiale f(0) =\n100.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Quelle est la puissance du son deux secondes après le\npincement de la corde ? Arrondir au watt près.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Résoudre par le calcul l’équation f(t) = 60, on donnera la\nvaleur exacte et la valeur approchée à 10\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8141em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan>\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mtight\">3\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span> .\nInterpréter ce résultat\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 15\u003C/strong>. \u003Cem>Un condensateur de capacité \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C\u003C/span>\u003C/span>\u003C/span>\u003C/span> farads est chargé sous une tension initiale de 20\nvolts.\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>Il se décharge ensuite dans un résistor de résistance \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\">R\u003C/span>\u003C/span>\u003C/span>\u003C/span> ohms. En notant \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">u\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">t\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span> la mesure de\nla tension en volts au bout de \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6151em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">t\u003C/span>\u003C/span>\u003C/span>\u003C/span> secondes aux bornes\ndu condensateur, \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">u\u003C/span>\u003C/span>\u003C/span>\u003C/span> est alors une fonction définie\nsur \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mopen\">[\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003Cspan class=\"mpunct\">;\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord\">+\u003C/span>\u003Cspan class=\"mord\">∞\u003C/span>\u003Cspan class=\"mopen\">[\u003C/span>\u003C/span>\u003C/span>\u003C/span>, qui est solution de l’équation\ndifférentielle : \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C/span>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">RC\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose nulldelimiter\">\u003C/span>\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/em>\u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Résoudre l’équation et en déduire la fonction \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">u\u003C/span>\u003C/span>\u003C/span>\u003C/span>.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Pour cette question, \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\">R\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">1000\u003C/span>\u003C/span>\u003C/span>\u003C/span> et \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8141em;\">\u003C/span>\u003Cspan class=\"mord\">1\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">0\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C/span>\u003Cspan class=\"mord mtight\">4\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>. Pendant combien de temps (au centième de\nseconde près) la tension aux bornes du condensateur reste-elle\nsupérieure ou égale à 5 volts ?\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\n\u003C/div>\n\u003Cp>\u003Cstrong>Résolution de problèmes\u003C/strong>\u003C/p>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 16\u003C/strong>. \u003Cem>Le nombre de bactéries \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">t\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span> d’une culture initialement à 600 passe au bout de 2\nheures à 1 800.\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>On suppose que le taux de croissance est proportionnel au nombre\nde bactéries présentes.\u003C/em>\u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Donner une équation différentielle qui traduit le problème\npuis déterminer \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">N\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">t\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span> à l’aide des conditions\nimposées.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Déterminer le nombre de bactéries après 4\nheures.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Déterminer le temps \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6151em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">t\u003C/span>\u003C/span>\u003C/span>\u003C/span> nécessaire pour que\nle nombre de bactéries dépasse 12 000.\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 17\u003C/strong>. \u003Cem>Monsieur Diop est promoteur d’une\nentreprise agricole. Il a acquis nouvellement une vaste terre plane sur\nlaquelle , il projette y produire de la papaye Il soumet son projet à un\nconseil d’ingénieurs pour une étude de marché afin de lui présenter les\natouts bénéficiaires sur ce produit.\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>Le conseil à la fin de cette étude basée sur un repère orthonormé\nd’unité graphique 1cm pour 100 m, adresse ses solutions à Pierre, un\nélève compétent en stage auprès du conseil, en ces termes :\u003C/em>\u003C/p>\n\u003Cp>\u003Cem><<Le bénéfice à réaliser en milliers de francs en fonction\nde la quantité \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003C/span>\u003C/span>\u003C/span> de papayes en tonnes par an est\ndonné par la fonction \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">h\u003C/span>\u003C/span>\u003C/span>\u003C/span> telle que \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0019em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">h\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">−\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0019em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord\">3\u003C/span>\u003Cspan class=\"mord mathnormal\">h\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">+\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord\">2\u003C/span>\u003Cspan class=\"mord mathnormal\">h\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord mathnormal\">x\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003C/span>\u003C/span>\u003C/span> et dont la\ncourbe intégrale passe par le point A(0;) et admet en ce point une\ntangente de coefficient directeur 10000.>>\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>\u003Cstrong>Tâche:\u003C/strong>\u003Cbr>\nDéterminer le bénéfice maximal annuel à réaliser par l’entreprise de\nmonsieur Diop s’il se lance dans la production de papayes.\u003C/em>\u003C/p>\n\u003C/div>\n\u003Cdiv class=\"exercice\">\n\u003Cp>\u003Cstrong>Exercice 18\u003C/strong>. \u003Cem>Le taux de croissance d’une\npopulation de girafes dans un parc national peut être modélisé par\nl’équation différentielle \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.2251em;vertical-align:-0.345em;\">\u003C/span>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C/span>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8801em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">d\u003C/span>\u003Cspan class=\"mord mathnormal mtight\">t\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C/span>\u003C/span>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C/span>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">d\u003C/span>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">P\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"vlist-s\">​\u003C/span>\u003C/span>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003C/span>\u003Cspan class=\"mclose nulldelimiter\">\u003C/span>\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003Cspan class=\"mrel\">=\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord\">0\u003C/span>\u003Cspan class=\"mpunct\">,\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C/span>\u003Cspan class=\"mord\">0002\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P\u003C/span>\u003Cspan class=\"mopen\">(\u003C/span>\u003Cspan class=\"mord\">200\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003Cspan class=\"mbin\">−\u003C/span>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C/span>\u003C/span>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P\u003C/span>\u003Cspan class=\"mclose\">)\u003C/span>\u003C/span>\u003C/span>\u003C/span> où \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6151em;\">\u003C/span>\u003Cspan class=\"mord mathnormal\">t\u003C/span>\u003C/span>\u003C/span>\u003C/span> est exprimé en\nannées.\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>On sait qu’il y avait 30 girafes quand les scientifiques\ncommençaient à étudier cette population.\u003C/em>\u003C/p>\n\u003Cp>\u003Cem>\u003Cstrong>Tâches\u003C/strong>\u003Cbr>\nDétermine:\u003C/em>\u003C/p>\n\u003Col>\n\u003Cli>\u003Cp>\u003Cem>Le nombre d’années qu’il faut pour cette population de\ngirafes atteignent 150.\u003C/em>\u003C/p>\u003C/li>\n\u003Cli>\u003Cp>\u003Cem>Quel maximum cette population peut-elle atteindre\n?\u003C/em>\u003C/p>\u003C/li>\n\u003C/ol>\n\u003C/div>\n","equa-diff","Équations différentielles","equations differentielles",[12],"algebra",1779628771704]