{"id":1540,"date":"2026-09-08T08:06:57","date_gmt":"2026-09-08T08:06:57","guid":{"rendered":"https:\/\/maroczain.com\/scolaire.maroczain.com\/?page_id=1540"},"modified":"2026-09-08T08:06:57","modified_gmt":"2026-09-08T08:06:57","slug":"modelise-resoudre-interpreter","status":"publish","type":"page","link":"https:\/\/maroczain.com\/scolaire.maroczain.com\/modelise-resoudre-interpreter\/","title":{"rendered":"Mod\u00e9lise. R\u00e9soudre. Interpr\u00e9ter."},"content":{"rendered":"\n&#8220;`html\n<div id=\"mza-ed-cpge\">\n\n<style>\n#mza-ed-cpge,#mza-ed-cpge *{box-sizing:border-box}\n#mza-ed-cpge{\n --navy:#06172c;--navy2:#0d416d;--gold:#c99a36;--gold2:#efd68c;\n --paper:#f5f6f8;--white:#fff;--text:#18283a;--muted:#687789;--line:#dfe5eb;\n font-family:Arial,Helvetica,sans-serif;background:var(--paper);color:var(--text);overflow:hidden\n}\n#mza-ed-cpge a{text-decoration:none;color:inherit}\n#mza-ed-cpge .wrap{max-width:1200px;margin:auto;padding:0 24px}\n#mza-ed-cpge section{padding:68px 0}\n#mza-ed-cpge .hero{\n color:#fff;padding:72px 0;\n background:\n radial-gradient(circle at 84% 15%,rgba(213,169,68,.28),transparent 28%),\n linear-gradient(135deg,#041224,#082b4c 65%,#10517e)\n}\n#mza-ed-cpge .badge{\n display:inline-block;padding:8px 13px;border-radius:999px;\n border:1px solid rgba(239,214,140,.4);color:var(--gold2);\n font-size:11px;font-weight:900;letter-spacing:1.3px\n}\n#mza-ed-cpge h1{font:500 clamp(42px,7vw,72px)\/1 Georgia,serif;margin:17px 0;color:#fff}\n#mza-ed-cpge .lead{max-width:880px;color:#d5e1ea;font-size:18px;line-height:1.75}\n#mza-ed-cpge .actions{display:flex;gap:9px;flex-wrap:wrap;margin-top:26px}\n#mza-ed-cpge .btn{\n min-height:47px;padding:0 18px;border:none;border-radius:10px;\n display:inline-flex;align-items:center;justify-content:center;cursor:pointer;\n font-size:12px;font-weight:900\n}\n#mza-ed-cpge .gold{background:linear-gradient(135deg,#bd892b,#efd483);color:#142238}\n#mza-ed-cpge .glass{background:rgba(255,255,255,.08);border:1px solid rgba(255,255,255,.18);color:#fff}\n#mza-ed-cpge .head{max-width:870px;margin-bottom:32px}\n#mza-ed-cpge .kicker{color:#987023;font-size:11px;font-weight:900;letter-spacing:1.3px}\n#mza-ed-cpge h2{color:var(--navy);font:500 clamp(31px,4vw,48px)\/1.1 Georgia,serif;margin:8px 0 11px}\n#mza-ed-cpge .head p{color:var(--muted);line-height:1.7}\n#mza-ed-cpge .grid{display:grid;grid-template-columns:repeat(2,1fr);gap:16px}\n#mza-ed-cpge .card{\n background:#fff;border:1px solid var(--line);border-radius:19px;padding:23px\n}\n#mza-ed-cpge .card h3{color:var(--navy);margin:0 0 10px;font-size:20px}\n#mza-ed-cpge .card p,#mza-ed-cpge .card li{color:var(--muted);line-height:1.7}\n#mza-ed-cpge .formula{\n margin:14px 0;padding:15px;border-left:4px solid var(--gold);\n background:#f8f6ef;border-radius:0 12px 12px 0;\n font-family:Georgia,serif;color:#213349;line-height:1.75\n}\n#mza-ed-cpge .tip{\n margin-top:14px;padding:13px;border-radius:11px;background:#eef3f7;\n color:#40546a;font-size:13px;line-height:1.6\n}\n#mza-ed-cpge .dark{background:var(--navy);color:#fff}\n#mza-ed-cpge .dark h2{color:#fff}\n#mza-ed-cpge .dark .head p{color:#c6d4df}\n#mza-ed-cpge .method{display:grid;grid-template-columns:repeat(6,1fr);gap:10px}\n#mza-ed-cpge .method div{\n padding:20px 10px;text-align:center;border-radius:14px;\n background:rgba(255,255,255,.07);border:1px solid rgba(255,255,255,.1)\n}\n#mza-ed-cpge .method b{display:block;color:var(--gold2);font-size:23px;margin-bottom:7px}\n#mza-ed-cpge .method span{font-size:10px;font-weight:900}\n#mza-ed-cpge .exercise{\n background:#fff;border:1px solid var(--line);border-radius:18px;padding:22px;margin-bottom:14px\n}\n#mza-ed-cpge .exercise h3{color:var(--navy);margin:0 0 10px}\n#mza-ed-cpge .level{\n display:inline-block;padding:6px 9px;margin-bottom:10px;border-radius:999px;\n background:#f1eee5;color:#7b5b1d;font-size:10px;font-weight:900\n}\n#mza-ed-cpge .exercise p{color:var(--muted);line-height:1.7}\n#mza-ed-cpge .hint{\n display:none;margin-top:12px;padding:15px;border-radius:12px;\n background:#eef3f7;color:#425469;line-height:1.65\n}\n#mza-ed-cpge .qcm{\n background:#fff;border:1px solid var(--line);border-radius:23px;padding:28px\n}\n#mza-ed-cpge .question{padding:20px;margin:15px 0;background:#f7f8fa;border-radius:14px}\n#mza-ed-cpge .question strong{display:block;color:var(--navy);margin-bottom:12px}\n#mza-ed-cpge label{display:block;padding:8px 0;color:#526174;cursor:pointer}\n#mza-ed-cpge #edResult{\n display:none;margin-top:18px;padding:20px;border-radius:14px;background:var(--navy);color:#fff\n}\n#mza-ed-cpge .cta{\n text-align:center;padding:50px 24px;border-radius:25px;\n background:linear-gradient(135deg,#06172c,#0d416d);color:#fff\n}\n#mza-ed-cpge .cta h2{color:#fff}\n#mza-ed-cpge .cta p{max-width:720px;margin:0 auto 22px;color:#cfdae4;line-height:1.7}\n@media(max-width:900px){\n #mza-ed-cpge .grid,#mza-ed-cpge .method{grid-template-columns:1fr}\n}\n<\/style>\n\n<header class=\"hero\">\n<div class=\"wrap\">\n\n<span class=\"badge\">COURS 09 \u2022 \u00c9QUATIONS DIFF\u00c9RENTIELLES<\/span>\n\n<h1>Mod\u00e9liser.<br>R\u00e9soudre. Interpr\u00e9ter.<\/h1>\n\n<p class=\"lead\">\nLes \u00e9quations diff\u00e9rentielles d\u00e9crivent l&#8217;\u00e9volution d&#8217;un syst\u00e8me \u00e0 partir\nde ses variations. Elles interviennent en m\u00e9canique, \u00e9lectricit\u00e9,\nthermodynamique, probabilit\u00e9s et mod\u00e9lisation scientifique.\n<\/p>\n\n<div class=\"actions\">\n<a href=\"#cours\" class=\"btn gold\">\ud83d\udcda COMMENCER<\/a>\n<a href=\"#methode\" class=\"btn glass\">\ud83e\udde0 M\u00c9THODE<\/a>\n<a href=\"#exercices\" class=\"btn glass\">\ud83e\udde9 EXERCICES<\/a>\n<a href=\"#qcm\" class=\"btn glass\">\u2753 QCM<\/a>\n<\/div>\n\n<\/div>\n<\/header>\n\n<section id=\"cours\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">1. PRINCIPE<\/span>\n<h2>Une inconnue qui est une fonction.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>\u00c9quation diff\u00e9rentielle<\/h3>\n\n<p>\nUne \u00e9quation diff\u00e9rentielle relie une fonction inconnue y\n\u00e0 une ou plusieurs de ses d\u00e9riv\u00e9es.\n<\/p>\n\n<div class=\"formula\">\ny&#8217; = 2y\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Ordre<\/h3>\n\n<p>\nL&#8217;ordre de l&#8217;\u00e9quation est l&#8217;ordre de la d\u00e9riv\u00e9e la plus \u00e9lev\u00e9e.\n<\/p>\n\n<div class=\"formula\">\ny&#8221; + y = 0\n<\/div>\n\n<p>est une \u00e9quation d&#8217;ordre 2.<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Solution<\/h3>\n\n<p>\nUne solution est une fonction qui v\u00e9rifie l&#8217;\u00e9quation sur un intervalle donn\u00e9.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Condition initiale<\/h3>\n\n<div class=\"formula\">\ny(0)=3\n<\/div>\n\n<p>\nElle permet souvent de s\u00e9lectionner une solution particuli\u00e8re dans une famille.\n<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">2. PREMIER ORDRE HOMOG\u00c8NE<\/span>\n<h2>Le mod\u00e8le exponentiel.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>Consid\u00e9rons :<\/p>\n\n<div class=\"formula\">\ny&#8217; = ay\n<\/div>\n\n<p>\nLa famille de solutions est :\n<\/p>\n\n<div class=\"formula\">\ny(x)=Ce<sup>ax<\/sup>\n<\/div>\n\n<p>o\u00f9 C est une constante r\u00e9elle.<\/p>\n\n<\/div>\n\n<div class=\"grid\" style=\"margin-top:16px\">\n\n<div class=\"card\">\n<h3>Si a &gt; 0<\/h3>\n<p>Les solutions non nulles ont une croissance exponentielle en valeur absolue.<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Si a &lt; 0<\/h3>\n<p>Les solutions tendent vers 0 lorsque x tend vers +\u221e.<\/p>\n<\/div>\n\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">3. PREMIER ORDRE AVEC SECOND MEMBRE<\/span>\n<h2>Solution homog\u00e8ne + solution particuli\u00e8re.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Forme<\/h3>\n\n<div class=\"formula\">\ny&#8217; + ay = b(x)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>\u00c9tape 1<\/h3>\n\n<p>R\u00e9soudre l&#8217;\u00e9quation homog\u00e8ne :<\/p>\n\n<div class=\"formula\">\ny&#8217; + ay = 0\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>\u00c9tape 2<\/h3>\n\n<p>Chercher une solution particuli\u00e8re y\u209a adapt\u00e9e \u00e0 b(x).<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Solution g\u00e9n\u00e9rale<\/h3>\n\n<div class=\"formula\">\ny = y\u2095 + y\u209a\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"methode\" class=\"dark\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\" style=\"color:#efd68c\">MZA DIFFERENTIAL ENGINE<\/span>\n<h2>Six r\u00e9flexes.<\/h2>\n<p>Identifier avant de r\u00e9soudre.<\/p>\n<\/div>\n\n<div class=\"method\">\n<div><b>01<\/b><span>ORDRE<\/span><\/div>\n<div><b>02<\/b><span>TYPE<\/span><\/div>\n<div><b>03<\/b><span>HOMOG\u00c8NE<\/span><\/div>\n<div><b>04<\/b><span>PARTICULI\u00c8RE<\/span><\/div>\n<div><b>05<\/b><span>CONDITIONS<\/span><\/div>\n<div><b>06<\/b><span>V\u00c9RIFIER<\/span><\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">4. \u00c9QUATION LIN\u00c9AIRE G\u00c9N\u00c9RALE DU PREMIER ORDRE<\/span>\n<h2>Facteur int\u00e9grant.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>\nConsid\u00e9rons :\n<\/p>\n\n<div class=\"formula\">\ny&#8217; + a(x)y = b(x)\n<\/div>\n\n<p>\nOn introduit un facteur int\u00e9grant de la forme :\n<\/p>\n\n<div class=\"formula\">\n\u03bc(x)=e<sup>\u222ba(x)dx<\/sup>\n<\/div>\n\n<p>\nafin de transformer le membre de gauche en d\u00e9riv\u00e9e d&#8217;un produit.\n<\/p>\n\n<div class=\"formula\">\n(\u03bcy)&#8217; = \u03bcb\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">5. SECOND ORDRE \u00c0 COEFFICIENTS CONSTANTS<\/span>\n<h2>Utiliser l&#8217;\u00e9quation caract\u00e9ristique.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>Consid\u00e9rons :<\/p>\n\n<div class=\"formula\">\nay&#8221; + by&#8217; + cy = 0\n<\/div>\n\n<p>\nOn associe l&#8217;\u00e9quation caract\u00e9ristique :\n<\/p>\n\n<div class=\"formula\">\nar\u00b2 + br + c = 0\n<\/div>\n\n<\/div>\n\n<div class=\"grid\" style=\"margin-top:16px\">\n\n<div class=\"card\">\n<h3>Deux racines r\u00e9elles distinctes<\/h3>\n\n<div class=\"formula\">\nr\u2081 \u2260 r\u2082\n<\/div>\n\n<p>Alors :<\/p>\n\n<div class=\"formula\">\ny=Ae<sup>r\u2081x<\/sup>+Be<sup>r\u2082x<\/sup>\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Racine double<\/h3>\n\n<div class=\"formula\">\nr\u2081=r\u2082=r\n<\/div>\n\n<p>Alors :<\/p>\n\n<div class=\"formula\">\ny=(A+Bx)e<sup>rx<\/sup>\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Racines complexes<\/h3>\n\n<p>Si :<\/p>\n\n<div class=\"formula\">\nr=\u03b1\u00b1i\u03b2\n<\/div>\n\n<p>alors :<\/p>\n\n<div class=\"formula\">\ny=e<sup>\u03b1x<\/sup>(A cos \u03b2x + B sin \u03b2x)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>R\u00e9flexe CPGE<\/h3>\n\n<div class=\"tip\">\nL&#8217;\u00e9quation caract\u00e9ristique doit devenir un automatisme d\u00e8s que les coefficients sont constants.\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">6. SECOND MEMBRE<\/span>\n<h2>Trouver une solution particuli\u00e8re adapt\u00e9e.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Second membre polynomial<\/h3>\n\n<p>\nOn essaie g\u00e9n\u00e9ralement un polyn\u00f4me de degr\u00e9 appropri\u00e9.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Second membre exponentiel<\/h3>\n\n<div class=\"formula\">\nb(x)=Ke<sup>\u03bbx<\/sup>\n<\/div>\n\n<p>\nOn tente une solution particuli\u00e8re de forme exponentielle adapt\u00e9e.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Second membre trigonom\u00e9trique<\/h3>\n\n<div class=\"formula\">\nA cos(\u03c9x)+B sin(\u03c9x)\n<\/div>\n\n<p>\nOn cherche une combinaison de sinus et cosinus.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>R\u00e9sonance<\/h3>\n\n<div class=\"tip\">\nSi la forme propos\u00e9e est d\u00e9j\u00e0 solution de l&#8217;\u00e9quation homog\u00e8ne,\nil faut modifier l&#8217;essai, souvent en le multipliant par x.\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">7. PROBL\u00c8ME DE CAUCHY<\/span>\n<h2>D\u00e9terminer les constantes.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Ordre 1<\/h3>\n\n<p>Une condition initiale typique :<\/p>\n\n<div class=\"formula\">\ny(x\u2080)=y\u2080\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Ordre 2<\/h3>\n\n<p>On rencontre g\u00e9n\u00e9ralement deux conditions :<\/p>\n\n<div class=\"formula\">\ny(x\u2080)=y\u2080<br>\ny'(x\u2080)=v\u2080\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Pourquoi ?<\/h3>\n\n<p>\nParce que la solution g\u00e9n\u00e9rale d&#8217;une \u00e9quation d&#8217;ordre 2 contient g\u00e9n\u00e9ralement deux constantes.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Interpr\u00e9tation<\/h3>\n\n<p>\nLes conditions initiales traduisent l&#8217;\u00e9tat du syst\u00e8me au d\u00e9but de l&#8217;\u00e9tude.\n<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">8. MOD\u00c9LISATION<\/span>\n<h2>Des math\u00e9matiques au r\u00e9el.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Refroidissement<\/h3>\n\n<div class=\"formula\">\nT&#8217; = -k(T-T\u2091)\n<\/div>\n\n<p>\nLa vitesse de variation de la temp\u00e9rature d\u00e9pend de l&#8217;\u00e9cart avec l&#8217;environnement.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Croissance<\/h3>\n\n<div class=\"formula\">\nN&#8217;=kN\n<\/div>\n\n<p>\nMod\u00e8le \u00e9l\u00e9mentaire de croissance exponentielle.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Circuit \u00e9lectrique<\/h3>\n\n<div class=\"formula\">\nL i&#8217; + Ri = E(t)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Oscillateur<\/h3>\n\n<div class=\"formula\">\nx&#8221; + \u03c9\u00b2x = 0\n<\/div>\n\n<p>\nMod\u00e8le fondamental des oscillations harmoniques.\n<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"exercices\" style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">9. EXERCICES PROGRESSIFS<\/span>\n<h2>Reconna\u00eetre le type d&#8217;\u00e9quation.<\/h2>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 1<\/span>\n\n<h3>Exercice 1 \u2014 Premier ordre<\/h3>\n\n<p>R\u00e9soudre :<\/p>\n\n<div class=\"formula\">\ny&#8217;=3y\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaEDHint('edHint1')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"edHint1\" class=\"hint\">\nReconnaissez imm\u00e9diatement le mod\u00e8le y&#8217;=ay.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 2<\/span>\n\n<h3>Exercice 2 \u2014 Condition initiale<\/h3>\n\n<p>R\u00e9soudre :<\/p>\n\n<div class=\"formula\">\ny&#8217;=-2y<br>\ny(0)=5\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaEDHint('edHint2')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"edHint2\" class=\"hint\">\nTrouvez d&#8217;abord la famille g\u00e9n\u00e9rale puis utilisez la condition initiale pour d\u00e9terminer la constante.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 3<\/span>\n\n<h3>Exercice 3 \u2014 Second ordre<\/h3>\n\n<p>R\u00e9soudre :<\/p>\n\n<div class=\"formula\">\ny&#8221; &#8211; 5y&#8217; + 6y = 0\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaEDHint('edHint3')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"edHint3\" class=\"hint\">\n\u00c9crivez l&#8217;\u00e9quation caract\u00e9ristique et factorisez-la.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU PR\u00c9PA<\/span>\n\n<h3>Exercice 4 \u2014 Oscillations<\/h3>\n\n<p>R\u00e9soudre :<\/p>\n\n<div class=\"formula\">\ny&#8221; + 4y = 0\n<\/div>\n\n<p>avec :<\/p>\n\n<div class=\"formula\">\ny(0)=1<br>\ny'(0)=0\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaEDHint('edHint4')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"edHint4\" class=\"hint\">\nL&#8217;\u00e9quation caract\u00e9ristique poss\u00e8de deux racines complexes conjugu\u00e9es purement imaginaires.\n<\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">10. PROBLEM LAB<\/span>\n<h2>Mission : refroidissement d&#8217;un syst\u00e8me.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<h3>Mission CPGE<\/h3>\n\n<p>\nUn objet de temp\u00e9rature T(t) est plac\u00e9 dans une pi\u00e8ce maintenue \u00e0 temp\u00e9rature constante T\u2091.\nOn mod\u00e9lise son \u00e9volution par :\n<\/p>\n\n<div class=\"formula\">\nT'(t) = -k(T(t)-T\u2091)\n<\/div>\n\n<p>Votre mission :<\/p>\n\n<ol>\n<li>introduire une nouvelle fonction u(t)=T(t)-T\u2091 ;<\/li>\n<li>obtenir une \u00e9quation diff\u00e9rentielle plus simple ;<\/li>\n<li>d\u00e9terminer la forme g\u00e9n\u00e9rale de T(t) ;<\/li>\n<li>utiliser une temp\u00e9rature initiale T(0)=T\u2080 ;<\/li>\n<li>\u00e9tudier la limite lorsque t tend vers +\u221e ;<\/li>\n<li>interpr\u00e9ter physiquement cette limite.<\/li>\n<\/ol>\n\n<div class=\"actions\">\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaEDHint('edMission1')\">\ud83d\udca1 INDICE<\/button>\n<button class=\"btn\" style=\"background:#06172c;color:#fff\" type=\"button\" onclick=\"mzaEDHint('edMission2')\">\ud83e\udde0 M\u00c9THODE<\/button>\n<\/div>\n\n<div id=\"edMission1\" class=\"hint\">\nLa translation u=T-T\u2091 transforme l&#8217;\u00e9quation en une \u00e9quation homog\u00e8ne du premier ordre.\n<\/div>\n\n<div id=\"edMission2\" class=\"hint\">\nR\u00e9solvez d&#8217;abord l&#8217;\u00e9quation sur u, r\u00e9introduisez T, puis appliquez la condition initiale.\nEnfin, \u00e9tudiez le terme exponentiel lorsque t devient grand.\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"qcm\" style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">11. QCM DE VALIDATION<\/span>\n<h2>Diagnostic \u00c9quations diff\u00e9rentielles.<\/h2>\n\n<p>Le score est affich\u00e9 sans montrer les r\u00e9ponses correctes.<\/p>\n<\/div>\n\n<div class=\"qcm\">\n\n<form id=\"edQuiz\">\n\n<div class=\"question\">\n<strong>1. L&#8217;\u00e9quation y&#8221;+y=0 est d&#8217;ordre :<\/strong>\n<label><input type=\"radio\" name=\"e1\" value=\"0\"> 1<\/label>\n<label><input type=\"radio\" name=\"e1\" value=\"1\"> 2<\/label>\n<label><input type=\"radio\" name=\"e1\" value=\"0\"> 3<\/label>\n<label><input type=\"radio\" name=\"e1\" value=\"0\"> 0<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>2. Les solutions de y&#8217;=ay sont de la forme :<\/strong>\n<label><input type=\"radio\" name=\"e2\" value=\"1\"> Ce<sup>ax<\/sup><\/label>\n<label><input type=\"radio\" name=\"e2\" value=\"0\"> Cx\u00b2<\/label>\n<label><input type=\"radio\" name=\"e2\" value=\"0\"> C\/x<\/label>\n<label><input type=\"radio\" name=\"e2\" value=\"0\"> ax+C uniquement<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>3. Pour une \u00e9quation lin\u00e9aire avec second membre, on cherche g\u00e9n\u00e9ralement :<\/strong>\n<label><input type=\"radio\" name=\"e3\" value=\"1\"> solution homog\u00e8ne + solution particuli\u00e8re<\/label>\n<label><input type=\"radio\" name=\"e3\" value=\"0\"> uniquement une constante<\/label>\n<label><input type=\"radio\" name=\"e3\" value=\"0\"> uniquement une primitive<\/label>\n<label><input type=\"radio\" name=\"e3\" value=\"0\"> une s\u00e9rie g\u00e9om\u00e9trique<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>4. Pour ay&#8221;+by&#8217;+cy=0, on utilise :<\/strong>\n<label><input type=\"radio\" name=\"e4\" value=\"1\"> l&#8217;\u00e9quation caract\u00e9ristique<\/label>\n<label><input type=\"radio\" name=\"e4\" value=\"0\"> uniquement une int\u00e9grale<\/label>\n<label><input type=\"radio\" name=\"e4\" value=\"0\"> le th\u00e9or\u00e8me de Pythagore<\/label>\n<label><input type=\"radio\" name=\"e4\" value=\"0\"> une s\u00e9rie harmonique<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>5. Une \u00e9quation d&#8217;ordre 2 poss\u00e8de g\u00e9n\u00e9ralement dans sa solution g\u00e9n\u00e9rale :<\/strong>\n<label><input type=\"radio\" name=\"e5\" value=\"1\"> deux constantes<\/label>\n<label><input type=\"radio\" name=\"e5\" value=\"0\"> aucune constante<\/label>\n<label><input type=\"radio\" name=\"e5\" value=\"0\"> une seule variable<\/label>\n<label><input type=\"radio\" name=\"e5\" value=\"0\"> toujours trois constantes<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>6. Les conditions initiales servent principalement \u00e0 :<\/strong>\n<label><input type=\"radio\" name=\"e6\" value=\"1\"> d\u00e9terminer les constantes de la solution<\/label>\n<label><input type=\"radio\" name=\"e6\" value=\"0\"> changer l&#8217;ordre de l&#8217;\u00e9quation<\/label>\n<label><input type=\"radio\" name=\"e6\" value=\"0\"> supprimer les d\u00e9riv\u00e9es<\/label>\n<label><input type=\"radio\" name=\"e6\" value=\"0\"> rendre toutes les solutions p\u00e9riodiques<\/label>\n<\/div>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaEDScore()\">\nVALIDER MON QCM\n<\/button>\n\n<\/form>\n\n<div id=\"edResult\"><\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"cta\">\n\n<div style=\"font-size:43px\">dy\/dx<\/div>\n\n<h2>R\u00e9soudre une \u00e9volution, pas seulement une \u00e9quation.<\/h2>\n\n<p>\nUne \u00e9quation diff\u00e9rentielle raconte comment un syst\u00e8me change.\nLe bon r\u00e9flexe consiste \u00e0 identifier son ordre, sa structure,\nr\u00e9soudre l&#8217;\u00e9quation homog\u00e8ne, traiter le second membre,\nappliquer les conditions puis interpr\u00e9ter la solution.\n<\/p>\n\n<div class=\"actions\" style=\"justify-content:center\">\n<a href=\"#cours\" class=\"btn gold\">\ud83d\udcda REVOIR LE COURS<\/a>\n<a href=\"#exercices\" class=\"btn glass\">\ud83e\udde9 EXERCICES<\/a>\n<a href=\"#qcm\" class=\"btn glass\">\u2753 REFAIRE LE QCM<\/a>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<script>\nfunction mzaEDHint(id){\n const el=document.getElementById(id);\n if(!el)return;\n el.style.display = el.style.display===\"block\" ? \"none\" : \"block\";\n}\n\nfunction mzaEDScore(){\n\n const form=document.getElementById(\"edQuiz\");\n const result=document.getElementById(\"edResult\");\n let score=0;\n let complete=true;\n\n [\"e1\",\"e2\",\"e3\",\"e4\",\"e5\",\"e6\"].forEach(function(name){\n   const answer=form.querySelector('input[name=\"'+name+'\"]:checked');\n\n   if(!answer){\n     complete=false;\n   }else{\n     score+=Number(answer.value);\n   }\n });\n\n result.style.display=\"block\";\n\n if(!complete){\n   result.innerHTML=\"<strong>R\u00e9pondez aux 6 questions avant de valider.<\/strong>\";\n   return;\n }\n\n const pct=Math.round(score\/6*100);\n\n let level=\"\u00c9quations diff\u00e9rentielles \u00e0 consolider\";\n if(pct>=50) level=\"Bases acquises\";\n if(pct>=67) level=\"Bon niveau\";\n if(pct>=84) level=\"Tr\u00e8s bonne ma\u00eetrise\";\n if(pct===100) level=\"Excellent niveau CPGE\";\n\n result.innerHTML=\n \"<strong style='font-size:30px;color:#efd68c'>\"+pct+\"%<\/strong>\"+\n \"<p><b>\"+level+\"<\/b><\/p>\"+\n \"<p>Score : \"+score+\" \/ 6<\/p>\"+\n \"<p>Les r\u00e9ponses correctes et la correction d\u00e9taill\u00e9e restent masqu\u00e9es.<\/p>\";\n}\n<\/script>\n\n<\/div>\n&#8220;`\n","protected":false},"excerpt":{"rendered":"<p>&#8220;`html COURS 09 \u2022 \u00c9QUATIONS DIFF\u00c9RENTIELLES Mod\u00e9liser.R\u00e9soudre. Interpr\u00e9ter. Les \u00e9quations diff\u00e9rentielles d\u00e9crivent l&#8217;\u00e9volution d&#8217;un syst\u00e8me \u00e0 partir de ses variations. Elles interviennent en m\u00e9canique, \u00e9lectricit\u00e9, thermodynamique, probabilit\u00e9s et mod\u00e9lisation scientifique. \ud83d\udcda COMMENCER \ud83e\udde0 M\u00c9THODE \ud83e\udde9 EXERCICES \u2753 QCM 1. PRINCIPE Une inconnue qui est une fonction. \u00c9quation diff\u00e9rentielle Une \u00e9quation diff\u00e9rentielle relie une fonction inconnue [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1540","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1540","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/comments?post=1540"}],"version-history":[{"count":1,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1540\/revisions"}],"predecessor-version":[{"id":1542,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1540\/revisions\/1542"}],"wp:attachment":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/media?parent=1540"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}