{"id":32822,"date":"2022-03-15T13:00:17","date_gmt":"2022-03-15T13:00:17","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32822"},"modified":"2025-04-03T13:30:02","modified_gmt":"2025-04-03T13:30:02","slug":"introduction-aux-equations-differentielles-ordinaires","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/introduction-aux-equations-differentielles-ordinaires\/","title":{"rendered":"Introduction aux \u00c9quations Diff\u00e9rentielles Ordinaires"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>Introduction aux \u00c9quations Diff\u00e9rentielles Ordinaires<\/h1>\n<p style=\"text-align:center;\"><em>Ce cours propose une exploration d\u00e9taill\u00e9e des id\u00e9es fondamentales qui r\u00e9gissent ces \u00e9quations et leurs applications dans divers domaines. En commen\u00e7ant par une analyse de la nature du changement incessant dans le monde qui nous entoure, on pr\u00e9sente des concepts de base tels que les fonctions, les d\u00e9riv\u00e9es et leur relation avec le changement continu et discret. On introduit la distinction entre les \u00c9quations Diff\u00e9rentielles Partielles (EDP) et les \u00c9quations Diff\u00e9rentielles Ordinaires (EDO), en se concentrant sur l\u2019\u00e9tude des EDO. Les concepts sont illustr\u00e9s par des exemples pratiques comme le refroidissement d\u2019une tasse de caf\u00e9, les lois de Newton et les mod\u00e8les de population. Les \u00e9tudiants auront l\u2019opportunit\u00e9 de se familiariser avec les \u00e9quations diff\u00e9rentielles qui r\u00e9gissent les ph\u00e9nom\u00e8nes naturels et physiques, de d\u00e9couvrir comment elles peuvent \u00eatre repr\u00e9sent\u00e9es math\u00e9matiquement et de comprendre certaines techniques permettant d\u2019\u00e9tudier leurs solutions. Ces connaissances initiales constitueront la base d\u2019\u00e9tudes plus avanc\u00e9es sur les \u00e9quations diff\u00e9rentielles et leurs applications en science et en ing\u00e9nierie.<\/em><\/p>\n<p style=\"text-align:center;\"><strong><u>Objectifs d&#8217;apprentissage<\/u> :<\/strong><br \/>\u00c0 l\u2019issue de ce cours, l\u2019\u00e9tudiant sera capable de :<\/p>\n<ol>\n<li><strong>Comprendre<\/strong> les concepts de base li\u00e9s aux \u00e9quations diff\u00e9rentielles, tels que la nature du changement, les fonctions, les d\u00e9riv\u00e9es et les diff\u00e9rences entre les \u00c9quations Diff\u00e9rentielles Partielles (EDP) et les \u00c9quations Diff\u00e9rentielles Ordinaires (EDO)<\/li>\n<\/ul>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/bYwm6NAEvVA\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><br \/>\n<center><\/p>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#LasEcuacionesDiferencialesYLaNaturalezaDeLasCosas\"><strong>Les \u00c9quations Diff\u00e9rentielles et la Nature des Choses<\/strong><\/a><br \/>\n<a href=\"#ElCambioIncesante\">Le Changement Incessant<\/a><br \/>\n<a href=\"#FuncionesDerivadasYSusCambios\">Fonctions, d\u00e9riv\u00e9es et leurs changements<\/a><br \/>\n<a href=\"#EDOyEDP\">EDO et EDP<\/a><br \/>\n<a href=\"#EjemplosDeEcuacionesDiferencialesOrdinarias\"><strong>Exemples d\u2019\u00c9quations Diff\u00e9rentielles Ordinaires<\/strong><\/a><br \/>\n<a href=\"#ElEnfriamientoDeUnaTazaDeCafe\">Le refroidissement d\u2019une tasse de caf\u00e9<\/a><br \/>\n<a href=\"#LasLeyesDeNewton\">Les Lois de Newton<\/a><br \/>\n<a href=\"#ModeloDePoblaciones\">Mod\u00e8le de populations<\/a>\n<\/p>\n<p><\/center><\/p>\n<p><a name=\"LasEcuacionesDiferencialesYLaNaturalezaDeLasCosas\"><\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/KgUDA2Q1qaA\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<h2>Les \u00c9quations Diff\u00e9rentielles et la Nature des Choses<\/h2>\n<p><a name=\"ElCambioIncesante\"><\/a><\/p>\n<h3>Le Changement Incessant<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=133s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Dans la nature, tout est en changement constant.<\/span><\/strong><\/a> M\u00eame ce qui semble ne jamais changer, comme la brillance du Soleil, varie si on l\u2019observe \u00e0 l\u2019\u00e9chelle temporelle appropri\u00e9e. Tout change : la brillance des \u00e9toiles, la temp\u00e9rature d\u2019une tasse de caf\u00e9, la position d\u2019un objet et la taille d\u2019une population en sont quelques exemples, et ces taux de changement sont g\u00e9n\u00e9ralement li\u00e9s \u00e0 l\u2019\u00e9tat de ce qui change pendant que ce changement se produit.<\/p>\n<p>Une mani\u00e8re intuitive de comprendre le changement est d\u2019observer comment les choses \u00e9voluent au fil du temps. Le changement qui se produit par rapport au temps est ce que nous appelons l\u2019\u00e9volution, et tout ce que nous pouvons observer est en \u00e9volution continue. Mais l\u2019\u00e9volution n\u2019est pas la seule forme de changement ; par exemple, bien que notre altitude par rapport au niveau de la mer puisse varier avec le temps, il est plus probable qu\u2019elle change selon notre position (ou coordonn\u00e9es g\u00e9ographiques).<\/p>\n<p><a name=\"FuncionesDerivadasYSusCambios\"><\/a><\/p>\n<h3>Fonctions, d\u00e9riv\u00e9es et leurs changements<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=301s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">En termes plus g\u00e9n\u00e9raux,<\/span><\/strong><\/a> une fonction de plusieurs variables <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_1,x_2, \\cdots, x_n)<\/span><\/span> peut varier si l\u2019une de ses variables change, et ce changement peut \u00eatre continu ou discret. Pour une fonction de plusieurs variables, le changement continu peut \u00eatre \u00e9tudi\u00e9 \u00e0 travers les <strong>d\u00e9riv\u00e9es partielles :<\/strong><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial f(x_1, \\cdots, x_n)}{\\partial x_1} = \\lim_{\\Delta x_1 \\to 0} \\frac{ f(x_1 + \\Delta x_1, \\cdots, x_n) -  f(x_1, \\cdots, x_n)}{\\Delta x_1} <\/span>\n<p>Si la fonction est d\u2019une seule variable, on utilise la <strong>d\u00e9riv\u00e9e ordinaire :<\/strong><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} = \\lim_{\\Delta x \\to 0} \\frac{ f(x + \\Delta x) -  f(x)}{\\Delta x} <\/span>\n<p>Si le changement est discret plut\u00f4t que continu, on omet simplement le calcul de la limite qui appara\u00eet dans les d\u00e9riv\u00e9es.<\/p>\n<p><a name=\"EDOyEDP\"><\/a><\/p>\n<h3>EDO et EDP<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=624s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Une \u00e9quation qui implique une fonction et ses diff\u00e9rentes d\u00e9riv\u00e9es<\/span><\/strong><\/a> est appel\u00e9e une <strong>\u00c9quation Diff\u00e9rentielle<\/strong>. Si ces d\u00e9riv\u00e9es sont partielles ou ordinaires, on parle respectivement d\u2019<strong>\u00c9quations Diff\u00e9rentielles Partielles (EDP)<\/strong> ou d\u2019<strong>\u00c9quations Diff\u00e9rentielles Ordinaires (EDO)<\/strong>. Pour l\u2019instant, nous allons nous concentrer sur l\u2019\u00e9tude des \u00e9quations diff\u00e9rentielles ordinaires et examiner quelques exemples dans lesquels elles apparaissent.<\/p>\n<p><a name=\"EjemplosDeEcuacionesDiferencialesOrdinarias\"><\/a><\/p>\n<h2>Exemples d\u2019\u00c9quations Diff\u00e9rentielles Ordinaires<\/h2>\n<p><a name=\"ElEnfriamientoDeUnaTazaDeCafe\"><\/a><\/p>\n<h3>Le refroidissement d\u2019une tasse de caf\u00e9<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=680s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Le taux de refroidissement d\u2019une tasse de caf\u00e9 est proportionnel<\/span><\/strong><\/a> \u00e0 la diff\u00e9rence de temp\u00e9rature entre l\u2019environnement et le caf\u00e9. Si la temp\u00e9rature de l\u2019air, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_a<\/span><\/span>, est constante et que la temp\u00e9rature du caf\u00e9 est une fonction du temps <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_c=T_c(t),<\/span><\/span>, on peut trouver une \u00e9quation diff\u00e9rentielle qui nous permettra de d\u00e9terminer la temp\u00e9rature du caf\u00e9 \u00e0 tout moment. Initialement, nous avons :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dT_c(t)}{dt} = -\\alpha^2(T_c(t) - T_a) <\/span>\n<p>O\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> est une constante de proportionnalit\u00e9, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_a \\lt T_c(t)<\/span><\/span> et le signe n\u00e9gatif indique que la temp\u00e9rature du caf\u00e9 diminue. Plus tard, nous verrons que cette \u00e9quation a une solution de la forme :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_c(t) = T_a + Be^{-\\alpha^2 t}<\/span>\n<p>O\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> est une constante \u00e0 d\u00e9terminer.<\/p>\n<p><a name=\"LasLeyesDeNewton\"><\/a><\/p>\n<h3>Les Lois de Newton<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=885s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">La deuxi\u00e8me loi de Newton est, essentiellement, une \u00e9quation diff\u00e9rentielle ordinaire,<\/span><\/strong><\/a> car dans l\u2019expression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F=ma<\/span><\/span> (force \u00e9gale \u00e0 masse multipli\u00e9e par acc\u00e9l\u00e9ration), l\u2019acc\u00e9l\u00e9ration, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=d^2x(t)\/dt^2,<\/span><\/span>, est la d\u00e9riv\u00e9e seconde de la position de l\u2019objet par rapport au temps. Gr\u00e2ce \u00e0 cette loi, nous pouvons trouver des relations qui d\u00e9crivent le mouvement des corps, qui sont en r\u00e9alit\u00e9 des \u00e9quations diff\u00e9rentielles. Un exemple simple est l\u2019\u00e9tude des ressorts : si nous avons un ressort fix\u00e9 \u00e0 un mur d\u2019un c\u00f4t\u00e9 et \u00e0 une masse de l\u2019autre, en position d\u2019\u00e9quilibre, et que nous d\u00e9pla\u00e7ons ensuite la masse d\u2019une distance <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> par rapport \u00e0 cette position, selon la loi de Hooke, la masse ressentira une force de rappel <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F=-kx<\/span><\/span>. Alors, selon la deuxi\u00e8me loi de Newton, nous aurons :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle -kx(t) = m\\frac{d^2x(t)}{dt^2} <\/span>\n<p>Nous verrons plus tard que sa solution est de la forme :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x(t) = A\\sin\\left(\\sqrt{\\frac{k}{m}}t + \\phi \\right)<\/span>\n<p>O\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> sont des constantes qui seront d\u00e9termin\u00e9es par les <strong>conditions initiales du probl\u00e8me<\/strong>.<\/p>\n<p><a name=\"ModeloDePoblaciones\"><\/a><\/p>\n<h3>Mod\u00e8le de populations<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=1184s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Le taux de croissance par habitant<\/span><\/strong><\/a> d\u2019une population est \u00e9gal \u00e0 la diff\u00e9rence entre les taux de natalit\u00e9 et de mortalit\u00e9, c\u2019est-\u00e0-dire :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{1}{x(t)} \\frac{dx(t)}{dt} = N - M<\/span>\n<p>Si le taux de natalit\u00e9 <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> reste constant dans le temps et que les d\u00e9c\u00e8s sont proportionnels \u00e0 la population, c\u2019est-\u00e0-dire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M=\\alpha^2 x(t),<\/span><\/span>, alors l\u2019\u00e9quation pr\u00e9c\u00e9dente prend la forme :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx(t)}{dt} = x(t) (N - \\alpha^2 x(t))<\/span>\n<p>Ceci est connu sous le nom d\u2019<strong>\u00ab\u00c9quation Logistique des Populations\u00bb<\/strong>. \u00c0 partir de cette \u00e9quation, on peut construire une g\u00e9n\u00e9ralisation pour de nombreuses populations <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1(t), x_2(t), \\cdots, x_n(t)<\/span><\/span> qui sont en concurrence pour exister, de la mani\u00e8re suivante :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx_i(t)}{dt} = x_i(t) \\left(N_i - \\displaystyle \\sum_{j=1}^n\\alpha^2_{ij} x_j(t)  \\right)<\/span>\n<p>Avec <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\in\\{1,\\cdots, n\\}<\/span><\/span>. C\u2019est ce que l\u2019on appelle les <strong>\u00c9quations de Lotka-Volterra<\/strong>.<\/p>\n<h2>Conclusion<\/h2>\n<p>Tout au long de cette introduction aux \u00c9quations Diff\u00e9rentielles Ordinaires, nous avons explor\u00e9 comment les math\u00e9matiques peuvent capturer de mani\u00e8re pr\u00e9cise et \u00e9l\u00e9gante les changements qui se produisent dans le monde naturel. Du refroidissement d\u2019une tasse de caf\u00e9 au mouvement d\u2019un ressort ou \u00e0 la croissance d\u2019une population, les EDO permettent de traduire des dynamiques complexes en relations math\u00e9matiques compr\u00e9hensibles et analysables.<\/p>\n<p>Comprendre la structure et le sens de ces \u00e9quations ouvre la porte \u00e0 de multiples disciplines comme la physique, la biologie, l\u2019\u00e9conomie et l\u2019ing\u00e9nierie. Ce cours pose les bases conceptuelles n\u00e9cessaires pour poursuivre des \u00e9tudes plus avanc\u00e9es, o\u00f9 seront approfondies les techniques de r\u00e9solution, l\u2019analyse qualitative et les m\u00e9thodes num\u00e9riques. Le plus important, cependant, est d\u2019avoir d\u00e9velopp\u00e9 une intuition initiale sur la mani\u00e8re dont le langage du changement \u2014 les \u00e9quations diff\u00e9rentielles \u2014 nous permet de d\u00e9crire, comprendre et pr\u00e9dire le comportement des syst\u00e8mes dynamiques.<\/p>\n<p>Dans les prochains cours, nous continuerons \u00e0 d\u00e9velopper des outils plus puissants et \u00e0 les appliquer \u00e0 de nouveaux contextes. Les \u00e9quations diff\u00e9rentielles ne nous offrent pas seulement un moyen d\u2019analyser la r\u00e9alit\u00e9, mais aussi d\u2019imaginer comment elle pourrait \u00e9voluer sous diff\u00e9rentes conditions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Introduction aux \u00c9quations Diff\u00e9rentielles Ordinaires Ce cours propose une exploration d\u00e9taill\u00e9e des id\u00e9es fondamentales qui r\u00e9gissent ces \u00e9quations et leurs applications dans divers domaines. En commen\u00e7ant par une analyse de la nature du changement incessant dans le monde qui nous entoure, on pr\u00e9sente des concepts de base tels que les fonctions, les d\u00e9riv\u00e9es et leur [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32792,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":15,"footnotes":""},"categories":[1162,569],"tags":[],"class_list":["post-32822","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-equations-differentielles-ordinaires","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Introduction aux \u00c9quations Diff\u00e9rentielles Ordinaires - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"D\u00e9couvrez comment les \u00e9quations diff\u00e9rentielles ordinaires expliquent des 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