From 2d8ec903dde29bbb908a3d80f9c23b487f1ea7de Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Wed, 26 May 2021 11:35:53 +0200 Subject: [PATCH] =?UTF-8?q?Feat:=20D=C3=A9but=20du=20chapitre=20sur=20les?= =?UTF-8?q?=20=C3=A9quations=20diff=C3=A9rentielles?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Complementaire/04_eq_diff/1B_eq_diff.pdf | Bin 0 -> 73626 bytes Complementaire/04_eq_diff/1B_eq_diff.tex | 78 ++++++++++++++++++ .../04_eq_diff/1E_pos_vitesse_acc.pdf | Bin 0 -> 61650 bytes .../04_eq_diff/1E_pos_vitesse_acc.tex | 21 +++++ Complementaire/04_eq_diff/2B_solutions.pdf | Bin 0 -> 100116 bytes Complementaire/04_eq_diff/2B_solutions.tex | 68 +++++++++++++++ Complementaire/04_eq_diff/exercises.tex | 29 +++++++ Complementaire/04_eq_diff/index.rst | 27 ++++++ 8 files changed, 223 insertions(+) create mode 100644 Complementaire/04_eq_diff/1B_eq_diff.pdf create mode 100644 Complementaire/04_eq_diff/1B_eq_diff.tex create mode 100644 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z0QL~N8xYj{YYw5ODRWw# zK8i@xJd?KW$&Gfk7Rk*sw=T?$XR+oS+9D8AR(=l<@GDDwgecSP3+xT;*=v1s literal 0 HcmV?d00001 diff --git a/Complementaire/04_eq_diff/1B_eq_diff.tex b/Complementaire/04_eq_diff/1B_eq_diff.tex new file mode 100644 index 0000000..414d31a --- /dev/null +++ b/Complementaire/04_eq_diff/1B_eq_diff.tex @@ -0,0 +1,78 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Équation differentielle - Cours} +\date{Mai 2021} + +\pagestyle{empty} + +\begin{document} + +\maketitle + +\section{Équation différentielle} + +\begin{definition} + +Une \textbf{équation différentielle} est une relation une variable ($x$, $t$...), une fonction ($f$) et les dérivées de cette fonction ($f'$, $f''$...). + +\textbf{Résoudre une équation différentielle} consiste à déterminer toutes les fonctions qui satisfont cette relation. + +\end{definition} + +\subsection*{Exemple} + +On souhaite résoudre l'équation différentielle $f'(x) = 3x^2$. + +Le cours sur la primitive nous permet résoudre cette équation. Une solution peut-être +\[ + f(x) = x^3 +\] +Mais il en existe d'autres +\[ + f(x) = x^3 + 1 \qquad \qquad + f(x) = x^3 + 2 \qquad \qquad + f(x) = x^3 - 1 \qquad \qquad + f(x) = x^3 - 4 \qquad \qquad +\] + +On peut vérifier que cette fonction est bien solution de cette équation la dérivant: + +\afaire{} + +On peut noter toutes ces solutions sous la forme suivantes +\[ + f(x) = x^3 + k \qquad \mbox{ avec } k \mbox{ un nombre réel} +\] +Cela signifie qu'il y a une infinité de solution à cette équation différentielle. Toutes les fonctions tracées dans le graphiques ci-dessous sont des solutions (et il en existe une infinité d'autres) + +\begin{center} + \begin{tikzpicture}[yscale=0.7, xscale=1.4] + \tkzInit[xmin=-3,xmax=3,xstep=1, + ymin=-5,ymax=5,ystep=1] + \tkzGrid + \tkzAxeXY[up space=0.5,right space=.5] + \tkzFct[domain = -3:3, line width=1pt]{x**3} + \tkzFct[domain = -3:3, line width=1pt]{x**3+1} + \tkzFct[domain = -3:3, line width=1pt]{x**3+2} + \tkzFct[domain = -3:3, line width=1pt]{x**3-1} + \tkzFct[domain = -3:3, line width=1pt]{x**3-4} + \end{tikzpicture} +\end{center} + +\begin{encadre}{ Notation } +Il y a différente façons de noter les dérivées dans les équations différentielles: + +\begin{multicols}{3} + Classique: $f'(x) = 3x^2$ + + Compacte: $y' = 3x^2$ + + Physicienne: $\dfrac{df}{dx} = 3x^2$ +\end{multicols} +~\\ +\end{encadre} + + +\end{document} diff --git a/Complementaire/04_eq_diff/1E_pos_vitesse_acc.pdf b/Complementaire/04_eq_diff/1E_pos_vitesse_acc.pdf new file mode 100644 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+\maketitle + +\setcounter{section}{1} +\section{Solutions d'équations différentielles} + +\begin{propriete}[équation $y' = a(x)$] + + Soit $a(x)$ une fonction réelle, on note $A(x)$ une primitive de $a(x)$. + + Alors les solutions de l'équation différentielle $y' = a(x)$ sont + \[ + f(x) = A(x) + k \mbox{ où } k \mbox{ est un nom réel} + \] +\end{propriete} + +\paragraph{Exemples}% +Les solutions de $y' = 10x + 1$ sont +\afaire{Donner 3 solutions de cette équation différentielle} + +\begin{propriete}[équation $y' = ay$] + + Soit $a$ un nombre réel non nul + + Alors les solutions de l'équation différentielle $y' = a y$ sont + \[ + f(x) = ke^{ax} \mbox{ où } k \mbox{ est un nom réel} + \] +\end{propriete} + +\paragraph{Exemples}% +Les solutions de $y' = 10y$ sont +\afaire{Donner 3 solutions de cette équation différentielle} + +\paragraph{Démonstration}% + +\envideo{https://video.opytex.org/videos/watch/d4233ed5-4e88-4be1-b470-58df67aefeb5}{Démonstration de la propriété} + + +\begin{propriete}[équation $y' = ay + b$] + + Soient $a$ et $b$ deux nombres réels non nuls + + Alors les solutions de l'équation différentielle $y' = a y + b$ sont + \[ + f(x) = ke^{ax} - \frac{b}{a} \mbox{ où } k \mbox{ est un nom réel} + \] +\end{propriete} + +\paragraph{Exemples}% +Les solutions de $y' = 10y + 5$ sont +\afaire{Donner 3 solutions de cette équation différentielle} + +\envideo{https://video.opytex.org/videos/watch/b6247c66-e834-46f9-adfa-af30cca4721}{Résoudre une équation du type $y' = ay + b$.} + + +\end{document} diff --git a/Complementaire/04_eq_diff/exercises.tex b/Complementaire/04_eq_diff/exercises.tex new file mode 100644 index 0000000..27a1bdf --- /dev/null +++ b/Complementaire/04_eq_diff/exercises.tex @@ -0,0 +1,29 @@ +\collectexercises{banque} +\begin{exercise}[subtitle={Position - vitesse - accélération}, step={1}, origin={Création}, topics={Équation différentielle}, tags={analyse, exponentiel, dérivation}] + \begin{enumerate} + \item On observe un mobile en mouvement et on décrit sa position verticale en fonction du temps $t$ en secondes par la fonction $z(t) = -4,9t^2 + 12$. + \begin{enumerate} + \item Déterminer la fonction décrivant la vitesse du module $v(t) = z'(t)$ (ou en notation physique $\dfrac{dz}{dt}$). + \item Déterminer la fonction décrivant l'accélération du module $a(t) = v'(t)$ (ou en notation physique $\dfrac{dv}{dt}$). + \item À quelle hauteur le mobile a été lâché? Quel était alors sa vitesse? Son accélération? + \end{enumerate} + \item On étudie un mobile en chute libre. On le lance à une hauteur de 10m au dessus du sol avec une vitesse de 1m/s. Un bilan des forces permet de connaître son accélération au cours du mouvement: $a(t) = -10$. + \begin{enumerate} + \item On rappelle que l'accélération est la dérivée de la vitesse ($a(t) = v'(t)$). Déterminer la fonction vitesse du mobile. + \item On rappelle que la vitesse est la dérivée de la position ($v(t) = z(t)$). Déterminer la fonction position du mobile. + \item Est-ce que les deux fonctions déterminées aux questions précédentes sont conformes aux conditions initiales? + \end{enumerate} + \item On considère, la fonction $m(t)$ qui modélise la masse d'une réactif dans une réaction chimique. Une étude cinétique de la réaction mène déduire que l'évolution de la masse du réactif (la vitesse de la réaction) est proportionnelle à cette masse du réactif. On traduit cela par la formule $\dfrac{dm}{dt} = -k \times m(t)$. + + Pour simplifier, on estimera que $k = 1$ et que l'on a donc $\dfrac{dm}{dt} = -m(t)$ + + \begin{enumerate} + \item Est-ce que $v(t)$ peut-être une fonction constante? + \item Est-ce que $v(t)$ peut-être une fonction polynôme? + \item Est-ce que $v(t)$ peut-être une fonction exponentielle? + \end{enumerate} + \end{enumerate} +\end{exercise} + + +\collectexercisesstop{banque} diff --git a/Complementaire/04_eq_diff/index.rst b/Complementaire/04_eq_diff/index.rst new file mode 100644 index 0000000..408b1dc --- /dev/null +++ b/Complementaire/04_eq_diff/index.rst @@ -0,0 +1,27 @@ +Équations différentielles +######################### + +:date: 2021-05-26 +:modified: 2021-05-26 +:authors: Benjamin Bertrand +:tags: Exponentielle, Équations différentielles +:category: Complementaire +:summary: Réintroduction des équations différentielles + +Étape 1: Position - vitesse - accélération +========================================== + +À partir d'une position, on cherche à retrouver la vitesse puis l'accélération. Puis on renverse le problème, à partir d'une accélération, on va chercher à retrouver une fonction décrivant la position. + +Le but est de passer assez vite sur la première partie pour se concentrer sur la 2e et introduire la 3e. + +.. image:: ./1E_pos_vitesse_acc.pdf + :height: 200px + :alt: Manipulation de position, vitesse et accélération. + +Bilan: Définition d'une équation différentielle avec en particulier les différentes formes à connaître. + +.. image:: ./1B_eq_diff.pdf + :height: 200px + :alt: Définition des éuqation différentielles +