From 1184f3256ac5509bf8ddd87d4a69be1db11cfae2 Mon Sep 17 00:00:00 2001
From: "flyingscorpio@arch-desktop" <tfranken@protonmail.com>
Date: Mon, 20 Sep 2021 09:57:35 +0200
Subject: [PATCH] Finish E5, equa diff 2nd ordre
---
analyse/exercices/main.tex | 73 ++++++++++++++++++++++++++++++++++++++
1 file changed, 73 insertions(+)
diff --git a/analyse/exercices/main.tex b/analyse/exercices/main.tex
index cfcb7d0..37849c1 100644
--- a/analyse/exercices/main.tex
+++ b/analyse/exercices/main.tex
@@ -206,6 +206,79 @@
\paragraph{$(E_5)$}
$y'' - 7y' + 10 y = (x + 3) e^{2x}$
+ \begin{enumerate}[label=\alph*)]
+
+ \item Solution homogène
+ \begin{align*}
+ r^2 - 7r + 10 = 0
+ \implies \Delta = 9
+ \implies
+ \left\{
+ \begin{array}{l}
+ r_1 = \frac{7 - 3}{2} = 2 \\\\
+ r_2 = \frac{7 + 3}{2} = 5 \\
+ \end{array}
+ \right.
+ \end{align*}
+ \begin{align*}
+ \implies
+ y_0 = \lambda e^{2x} + \mu e^{5x}
+ \end{align*}
+
+ \item Solution particulière
+
+ Second membre~: $(x + 3)e^{\alpha x}$ avec $\alpha = 2$ $\implies \alpha$ racine de l'équation caractéristique.
+ \begin{align*}
+ &\left\{
+ \begin{array}{l}
+ y_1 = xe^{2x} (ax + b) \\
+ y_1' = (2x + 1)e^{2x}(ax + b) + axe^{2x} \\
+ y_1'' = (4ax + 2a + 2b)e^{2x} + (4ax^2 + 4ax + 4bx + 2b)e^{2x} \\
+ \end{array}
+ \right. \\
+ \implies
+ &\left\{
+ \begin{array}{l}
+ y_1 = xe^{2x} (ax + b) \\
+ y_1' = (2ax^2 + 2ax + 2bx + b)e^{2x} \\
+ y_1'' = (4ax^2 + 8ax + 4bx + 2a + 4b)e^{2x} \\
+ \end{array}
+ \right.
+ \end{align*}
+
+ Dans $(E_5)$~:
+
+ $(4ax^2 + 8ax + 4bx + 2a + 4b)e^{2x} - 7(2ax^2 + 2ax + 2bx + b)e^{2x} + 10(ax + b)xe^{2x} = (x + 3) e^{2x}$
+ \begin{align*}
+ \implies
+ -6ax + 2a - 3b = x + 3
+ \implies
+ \left\{
+ \begin{array}{l}
+ -6a = 1 \\
+ 2a - 3b = 3 \\
+ \end{array}
+ \right.
+ \implies
+ \left\{
+ \begin{array}{l}
+ a = \frac{-1}{6} \\\\
+ b = \frac{-10}{9} \\
+ \end{array}
+ \right.
+ \end{align*}
+ \begin{align*}
+ \implies
+ y_1 = xe^{2x}(-\frac{1}{6}x - \frac{10}{9})
+ \end{align*}
+
+ \item Solution générale
+ \begin{equation*}
+ y = y_0 + y_1 = \boxed{\lambda e^{2x} + \mu e^{5x} + xe^{2x}(-\frac{1}{6}x - \frac{10}{9})}
+ \end{equation*}
+
+ \end{enumerate}
+
\paragraph{$(E_6)$}
$y'' - y = x^3$