{"id":922,"date":"2018-12-27T18:14:28","date_gmt":"2018-12-27T17:14:28","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=922"},"modified":"2018-12-27T18:14:31","modified_gmt":"2018-12-27T17:14:31","slug":"introduction-aux-equations-differentielles","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/","title":{"rendered":"Introduction aux \u00e9quations diff\u00e9rentielles"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Introduction\" >Introduction<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Definition\" >D\u00e9finition<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Exemples_dequations_differentielles_lineaires\" >Exemples d&rsquo;\u00e9quations diff\u00e9rentielles lin\u00e9aires<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Resolution_de_y%E2%80%98_ay\" >R\u00e9solution de y&lsquo; = ay<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Exemple\" >Exemple<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Resolution_de_y%E2%80%98_ay_fx\" >R\u00e9solution de y&lsquo; + ay = f(x)<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Exemple_y%E2%80%98_%E2%80%93_3y_x%C2%B2\" >Exemple : y&lsquo; &#8211; 3y\u00a0=\u00a0x\u00b2<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Equation_de_la_forme_ay_%C2%BB_by_c_0\" >\u00c9quation de la forme ay\u00a0\u00bb + by + c = 0<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Equation_caracteristique\" >\u00c9quation caract\u00e9ristique<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Solutions\" >Solutions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Un_cas_particulier_lequation_y_%C2%BBomega2y0\" >Un cas particulier : l&rsquo;\u00e9quation \\(y\u00a0\u00bb+\\omega^2y=0\\)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Bonus_avec_second_membre\" >Bonus : avec second membre<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Les_applications_des_equations_differentielles\" >Les applications des \u00e9quations diff\u00e9rentielles<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Loi_de_Malthus\" >Loi de Malthus<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Dans_un_circuit_electrique\" >Dans un circuit \u00e9lectrique<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/#Oscillation_mecanique\" >Oscillation m\u00e9canique<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Introduction\"><\/span>Introduction<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">La r\u00e9forme du lyc\u00e9e s&rsquo;accompagne de son lot de nouveaut\u00e9s. En math\u00e9matiques, des bruits courent sur la r\u00e9apparition des \u00e9quations diff\u00e9rentielles dans le programme de la classe de maturit\u00e9 (Terminale).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ceci me donne l&rsquo;occasion de faire une br\u00e8ve (?) introduction de cette notion.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Definition\"><\/span>D\u00e9finition<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Dans <em>\u00e9quation\u00a0diff\u00e9rentielle<\/em>,\u00a0il y a d&rsquo;abord \u00ab\u00a0\u00e9quation\u00a0\u00bb. Par cons\u00e9quent, il va y avoir au moins une inconnue (il n&rsquo;y en a qu&rsquo;une seule pour commencer). Cette inconnue, c&rsquo;est une fonction. Elle est souvent d\u00e9sign\u00e9e par la lettre <em>y<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ensuite, il y a le mot \u00ab\u00a0diff\u00e9rentielle\u00a0\u00bb, ce qui signifie que notre fonction inconnue va appara\u00eetre sous diverses formes dans notre \u00e9quation en termes de diff\u00e9rentiation, ce qui veut dire qu&rsquo;il peut y avoir la fonction elle-m\u00eame (<em>y<\/em>), mais aussi sa d\u00e9riv\u00e9e (<em>y&rsquo;<\/em>) ainsi que sa d\u00e9riv\u00e9e seconde (<em>y\u00a0\u00bb<\/em>), etc. Au lyc\u00e9e, on s&rsquo;arr\u00eate \u00e0 la d\u00e9riv\u00e9e seconde.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si <em>y\u00a0\u00bb<\/em> appara\u00eet dans l&rsquo;\u00e9quation, on dira que l&rsquo;on a affaire \u00e0 une \u00e9quation diff\u00e9rentielle d&rsquo;<strong>ordre 2<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si tel n&rsquo;est pas le cas mais que <em>y&rsquo;<\/em> appara\u00eet, on dire que l&rsquo;\u00e9quation est d&rsquo;<strong>ordre 1.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Les coefficients de <em>y<\/em>, <em>y&rsquo;<\/em> et <em>y\u00a0\u00bb<\/em> sont au lyc\u00e9e des constantes, mais ils peuvent aussi \u00eatre des fonctions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">De plus, les \u00e9quations diff\u00e9rentielles seront <em>lin\u00e9aires<\/em>, c&rsquo;est-\u00e0-dire qu&rsquo;il n&rsquo;y aura pas d&rsquo;exposant aux inconnues (donc, par exemple, pas de <em>y<\/em>\u00b2).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Exemples_dequations_differentielles_lineaires\"><\/span>Exemples d&rsquo;\u00e9quations diff\u00e9rentielles lin\u00e9aires<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"wp-block-list\"><li><em><strong>y&rsquo;<\/strong><\/em><strong> + 3<\/strong><em><strong>y<\/strong><\/em><strong> = <\/strong><em><strong>x<\/strong><\/em><strong>\u00b2<\/strong> est une \u00e9quation diff\u00e9rentielle d&rsquo;ordre 1 \u00e0 coefficients constants. R\u00e9soudre cette \u00e9quation revient \u00e0 trouver toutes les fonctions <em>y<\/em> telles que <em>y&rsquo;<\/em>(<em>x<\/em>)+3<em>y<\/em>(<em>x<\/em>)=<em>x<\/em>\u00b2 pour tout r\u00e9el <em>x<\/em>.<\/li><li><em><strong>y<\/strong><\/em><strong>\u00a0\u00bb + 3<\/strong><em><strong>y<\/strong><\/em><strong>&lsquo; &#8211; 5<\/strong><em><strong>y<\/strong><\/em><strong> = 0<\/strong> est une \u00e9quation diff\u00e9rentielle d&rsquo;ordre 2 \u00e0 coefficients constants.<\/li><li><strong>(<\/strong><em><strong>x<\/strong><\/em><strong>+1)<\/strong><em><strong>y<\/strong><\/em><strong>\u00a0\u00bb &#8211; <\/strong><em><strong>x\u00b2y\u00a0= x<\/strong><\/em><strong>\u00b2 + <\/strong><em><strong>x<\/strong><\/em><strong> + 1<\/strong> est une \u00e9quation diff\u00e9rentielle d&rsquo;ordre 2 \u00e0 coefficients non constants.<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Resolution_de_y%E2%80%98_ay\"><\/span>R\u00e9solution de <em>y<\/em>&lsquo; = <em>ay<\/em><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Nous allons avant tout nous pencher sur cette \u00e9quation, qui est la plus simple (ou presque).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Il ne faut jamais oublier que <em>y<\/em> repr\u00e9sente une fonction (ce que l&rsquo;on oublie assez souvent au d\u00e9but car nous avions affaire jusqu&rsquo;\u00e0 pr\u00e9sent qu&rsquo;\u00e0 des \u00e9quations dont les inconnues \u00e9taient des nombres).$$\\begin{align}y&rsquo;=ay &amp; \\iff \\frac{y&rsquo;}{y}=a\\\\ &amp; \\iff \\int_{\\mathbb{R}}\\frac{y&rsquo;}{y}\\text{d}x=\\int_{\\mathbb{R}} a\\text{d}x\\\\&amp;\\iff \\ln\\big|y(x)\\big|=ax+k,\\ k\\in\\mathbb{R}\\\\&amp;\\iff y(x)=\\text{e}^{ax+k}=\\text{e}^k\\times\\text{e}^{ax}\\\\&amp;\\iff y(x)=C\\text{e}^{ax},\\ c\\in\\mathbb{R}.\\end{align}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pr\u00e9cisions : \u00e0 la deuxi\u00e8me ligne, on cherche les primitives de \\(\\frac{y&rsquo;}{y}\\), qui est de la forme \\(\\frac{u&rsquo;}{u}\\), et les primitives de \\(\\frac{u&rsquo;}{u}\\) sont les fonctions \\(\\ln|u|\\).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Exemple\"><\/span>Exemple<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">L&rsquo;\u00e9quation \\( y&rsquo;=-5y\\) admet pour solutions les fonctions: $$y(x)=C\\text{e}^{5x},\\ C\\in\\mathbb{R}.$$Il y a donc une infinit\u00e9 de solutions.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Resolution_de_y%E2%80%98_ay_fx\"><\/span>R\u00e9solution de <em>y<\/em>&lsquo; + <em>ay<\/em> = <em>f<\/em>(<em>x<\/em>)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Cette \u00e9quation se r\u00e9sout en deux temps.<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>D&rsquo;abord, on r\u00e9sout l&rsquo;\u00e9quation sans second membre <em>y<\/em>&lsquo; + <em>ay<\/em> = 0 \u00e0 l&rsquo;aide de ce que l&rsquo;on a dit pr\u00e9c\u00e9demment; cette \u00e9quation admet pour solutions les fonctions \\(y_0(x)=C\\text{e}^{-ax},\\ C\\in\\mathbb{R}\\).<\/li><li>Ensuite on trouve une solution particuli\u00e8re, c&rsquo;est-\u00e0-dire une fonction \\(y_p\\) telle que \\(y_p^\\prime(x)+ay_p(x)=f(x)\\).<\/li><li>L&rsquo;ensemble des solutions de l&rsquo;\u00e9quation diff\u00e9rentielle sera l&rsquo;ensemble des fonctions \\(y(x)=y_0(x)+y_p(x)\\).<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Exemple_y%E2%80%98_%E2%80%93_3y_x%C2%B2\"><\/span>Exemple : <em>y<\/em>&lsquo; &#8211; 3<em>y\u00a0=\u00a0x<\/em>\u00b2<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">L&rsquo;\u00e9quation homog\u00e8ne associ\u00e9e \u00e0 cette \u00e9quation est <em>y<\/em>&lsquo; &#8211; 3<em>y<\/em> = 0, admettant \\(y_0(x)=C\\text{e}^{3x}\\, C\\in\\mathbb{R}\\) comme solutions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">De plus, le second membre de l&rsquo;\u00e9quation diff\u00e9rentielle \u00e9tant un polyn\u00f4me de degr\u00e9 2, une solution particuli\u00e8re peut \u00eatre un polyn\u00f4me de degr\u00e9 3 (car une fois d\u00e9riv\u00e9, cela donnera un polyn\u00f4me de degr\u00e9 2).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Posons alors \\(y_p(x)=ax^3+bx^2+cx+d\\); ainsi, \\(y_p^\\prime(x)=3ax^2+2bx+c\\). Donc:$$\\begin{align} &amp; y_p^\\prime(x)-3y_p(x)=x^2\\\\\\iff&amp; 3ax^2+2bx+c-3(ax^3+bx^2+cx+d)=x^2\\\\\\iff&amp; -3ax^3+(3a-3b)x^2+(2b-3c)x+c-3d=x^2\\\\\\iff&amp;\\begin{cases} -3a=0\\\\3a-3b=1\\\\ 2b-3c=0\\\\c-3d=0\\end{cases}\\\\\\iff&amp;a=0,\\ b=-\\frac{1}{3},\\ c=-\\frac{2}{9},\\ d=-\\frac{2}{27}.\\end{align}$$Donc \\(y_p(x)=-\\frac{1}{3}x^2-\\frac{2}{9}x-\\frac{2}{27}\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, les solutions de l&rsquo;\u00e9quation diff\u00e9rentielle initiale sont:$$y(x)= <br>-\\frac{1}{3}x^2-\\frac{2}{9}x-\\frac{2}{27} + C\\text{e}^{3x}\\, C\\in\\mathbb{R}.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Equation_de_la_forme_ay_%C2%BB_by_c_0\"><\/span>\u00c9quation de la forme <em>ay<\/em>\u00a0\u00bb + <em>by<\/em> + <em>c<\/em> = 0<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Equation_caracteristique\"><\/span>\u00c9quation caract\u00e9ristique<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Ce genre d&rsquo;\u00e9quation diff\u00e9rentielle est souvent accompagn\u00e9 de son <em>\u00e9quation\u00a0caract\u00e9ristique<\/em>. Pour obtenir l&rsquo;\u00e9quation caract\u00e9ristique, on remplace <em>y<\/em>\u00a0\u00bb par <em>r<\/em>\u00b2 et <em>y<\/em> par <em>r<\/em>; on obtient alors <em>ar<\/em>\u00b2+<em>br<\/em>+<em>c<\/em>=0, \u00e9quation du second degr\u00e9 bien connue. On arrive alors \u00e0 d\u00e9montrer que les solutions de l&rsquo;\u00e9quation diff\u00e9rentielles sont de la forme \\( A\\text{e}^{r_1x} +B\\text{e}^{r_2x}\\), o\u00f9 \\(r_1\\) et \\(r_2\\) sont les solutions (r\u00e9elles ou complexes) de l&rsquo;\u00e9quation caract\u00e9ristique.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On obtient ainsi le th\u00e9or\u00e8me suivant.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Solutions\"><\/span>Solutions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<ul class=\"wp-block-list\"><li>si \\(\\Delta=b^2-4ac&lt;0\\) alors les solutions sont:$$y(x)=\\big[A\\cos(\\beta x)+B\\sin(\\beta x)\\big]\\text{e}^{\\alpha x},\\ A,B\\in\\mathbb{R}$$o\u00f9 \\(\\alpha=-\\frac{b}{2a}\\) et \\(\\beta=\\frac{\\sqrt{|\\Delta|}}{2a}\\).<\/li><li>si \\(\\Delta=0\\) alors les solutions sont:$$y(x)=(Ax+B)\\text{e}^{rx},\\ A,B\\in\\mathbb{R}$$o\u00f9 \\(r=-\\frac{b}{2a}\\).<\/li><li>si \\(\\Delta>0\\) alors les solutions sont:$$y(x)=A\\text{e}^{r_1x}+B\\text{e}^{r_2x},\\ A,B\\in\\mathbb{R}$$o\u00f9 \\(r_1\\) et \\(r_2\\) sont les racines du polyn\u00f4me <em>ax<\/em>\u00b2+<em>bx<\/em>+<em>c<\/em>.<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Un_cas_particulier_lequation_y_%C2%BBomega2y0\"><\/span>Un cas particulier : l&rsquo;\u00e9quation \\(y\u00a0\u00bb+\\omega^2y=0\\)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Les solutions sont, d&rsquo;apr\u00e8s le th\u00e9or\u00e8me vu pr\u00e9c\u00e9demment:$$y(x)=A\\cos(\\omega x)+B\\sin(\\omega x).$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Bonus_avec_second_membre\"><\/span>Bonus : avec second membre<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Si l&rsquo;\u00e9quation est de la forme <em>ay<\/em>\u00a0\u00bb + <em>by<\/em> + <em>c<\/em> = <em>f<\/em>(<em>x<\/em>) alors pour la r\u00e9soudre, on fera la m\u00eame chose que pour les \u00e9quations du premier ordre : somme des solutions de l&rsquo;\u00e9quation homog\u00e8ne associ\u00e9e et d&rsquo;une solution particuli\u00e8re.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Les_applications_des_equations_differentielles\"><\/span>Les applications des \u00e9quations diff\u00e9rentielles<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Loi_de_Malthus\"><\/span>Loi de Malthus<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Consid\u00e9rons une culture de bact\u00e9ries en milieu clos. La loi de Malthus est un mod\u00e8le d&rsquo;\u00e9volution disant que la vitesse d&rsquo;accroissement des bact\u00e9ries est proportionnelle au nombre de bact\u00e9ries pr\u00e9sentes. Ainsi,  <em>y<\/em>&lsquo; = <em>ay, <\/em>o\u00f9 <em>y(t)<\/em> repr\u00e9sente le nombre de bact\u00e9ries \u00e0 l&rsquo;instant <em>t<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Dans la pratique, ce mod\u00e8le n&rsquo;est pas vraisemblable. On lui pr\u00e9f\u00e9rera le mod\u00e8le de Verhulst:$$y&rsquo;=ay(M-y)$$mais cette \u00e9quation n&rsquo;\u00e9tant pas lin\u00e9aire, je n&rsquo;en  parlerai pas ici (peut-\u00eatre dans un autre article).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Dans_un_circuit_electrique\"><\/span>Dans un circuit \u00e9lectrique<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"328\" height=\"160\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/12\/circuit-electrique.png\" alt=\"\" class=\"wp-image-930\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/12\/circuit-electrique.png 328w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/12\/circuit-electrique-300x146.png 300w\" sizes=\"auto, (max-width: 328px) 100vw, 328px\" \/><figcaption>Repr\u00e9sentation d&rsquo;un circuit \u00e9lectrique<\/figcaption><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">On consid\u00e8re un circuit \u00e9lectrique o\u00f9  u(t) est la tension \u00e9lectrique aux bornes d\u2019un condensateur C aliment\u00e9 \u00e0 travers une r\u00e9sistance R sous une tension constante E.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> Les lois de l\u2019\u00e9lectricit\u00e9 indiquent que:$$RCu&rsquo;+u=E.$$Ainsi, pour trouver la tension, on doit r\u00e9soudre une \u00e9quation diff\u00e9rentielle.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Oscillation_mecanique\"><\/span>Oscillation m\u00e9canique<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"220\" height=\"160\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/12\/oscillation-mecanique.png\" alt=\"\" class=\"wp-image-931\"\/><figcaption>Oscillation m\u00e9canique et \u00e9quation diff\u00e9rentielle<\/figcaption><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Consid\u00e9rons une masse <em>m<\/em> suspendue \u00e0 un ressort de constante de raideur <em>k<\/em>. <em>x<\/em> d\u00e9signe la position de la masse par rapport \u00e0 sa position d\u2019\u00e9quilibre. Le frottement est suppos\u00e9 proportionnel \u00e0 la vitesse <em>v<\/em> = <em>x&rsquo;<\/em>(<em>t<\/em>). \u03bb est le coefficient de frottement (\u03bb>0).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Les lois de la m\u00e9canique du mouvement nous indiquent que:$$m\u22c5x\u2019\u2019(t) + \u03bb\u22c5x\u2019(t) + k\u22c5x(t) = 0.$$Ainsi, pour trouver la position de la masse, il faut r\u00e9soudre une \u00e9quation diff\u00e9rentielle. <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Introduction La r\u00e9forme du lyc\u00e9e s&rsquo;accompagne de son lot de nouveaut\u00e9s. En math\u00e9matiques, des bruits courent sur la r\u00e9apparition des \u00e9quations diff\u00e9rentielles dans le programme de la classe de maturit\u00e9 (Terminale). Ceci me donne l&rsquo;occasion de faire une br\u00e8ve (?) introduction de cette notion. D\u00e9finition Dans \u00e9quation\u00a0diff\u00e9rentielle,\u00a0il y a d&rsquo;abord [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[83],"class_list":["post-922","post","type-post","status-publish","format-standard","hentry","category-mathematiques","tag-equations-differentielles"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Introduction aux \u00e9quations diff\u00e9rentielles - Mathweb.fr<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/12\/27\/introduction-aux-equations-differentielles\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Introduction aux \u00e9quations diff\u00e9rentielles - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"Introduction La r\u00e9forme du lyc\u00e9e s&rsquo;accompagne de son lot de nouveaut\u00e9s. En math\u00e9matiques, des bruits courent sur la r\u00e9apparition des \u00e9quations diff\u00e9rentielles dans le programme de la classe de maturit\u00e9 (Terminale). Ceci me donne l&rsquo;occasion de faire une br\u00e8ve (?) introduction de cette notion. 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