From cb2ed80e938ae1e8676b395763a8f3ebb73fa0b2 Mon Sep 17 00:00:00 2001
From: "flyingscorpio@pinebookpro" <tfranken@protonmail.com>
Date: Tue, 21 Sep 2021 15:54:45 +0200
Subject: [PATCH] Add E7 equa diff 2nd ordre

---
 analyse/exercices/main.tex | 86 +++++++++++++++++++++++++++++++++++++-
 1 file changed, 85 insertions(+), 1 deletion(-)

diff --git a/analyse/exercices/main.tex b/analyse/exercices/main.tex
index 18e0b34..961c3dc 100644
--- a/analyse/exercices/main.tex
+++ b/analyse/exercices/main.tex
@@ -349,12 +349,96 @@
                     y = y_0 + y_1 = \boxed{\lambda e^{-x} + \mu e^{x} - x^3 - 6x}
                 \end{equation*}
 
-
         \end{enumerate}
 
     \paragraph{$(E_7)$}
         $y'' + y = \cos{x}$
 
+        \begin{enumerate}[label=\alph*)]
+
+            \item Solution homogène
+                \begin{align*}
+                    r^2 + 1 = 0
+                    \implies \Delta = -4
+                    \implies
+                    \left\{
+                    \begin{array}{l}
+                        r_1 = \frac{0 - 2i}{2} \\\\
+                        r_2 = \frac{0 + 2i}{2} \\
+                    \end{array}
+                    \right.
+                    \implies
+                    \left\{
+                    \begin{array}{l}
+                        \alpha = 0 \\
+                        \beta = 1
+                    \end{array}
+                    \right.
+                \end{align*}
+                \begin{align*}
+                    \implies
+                    y_0 = e^{0}(\lambda \cos{x} + \mu \sin{x})
+                    = \lambda \cos{x} + \mu \sin{x}
+                \end{align*}
+
+            \item Solution particulière
+
+                second membre~: $e^{\alpha x}\cos{x}$ avec $\alpha = 0$ $\implies \alpha$ non racine de l'équation caractéristique.
+                \begin{align*}
+                    y_1 &= xe^{\alpha x}(a\cos{\beta x} + b\sin{\beta x}) \\
+                        &= x(a\cos{x} + b\sin{x}) \\
+                    \implies
+                    &\left\{
+                    \begin{array}{l}
+                        y_1 = x(a\cos{x} + b\sin{x}) \\
+                        y_1' = a\cos{x} + b\sin{x} - x(a\sin{x} - b\cos{x}) \\
+                        y_1'' = -a\sin{x} + b\cos{x} - a\sin{x} + b\cos{x} - x(a\cos{x} + b\sin{x}) \\
+                    \end{array}
+                    \right. \\
+                    \implies
+                    &\left\{
+                    \begin{array}{l}
+                        y_1 = x(a\cos{x} + b\sin{x}) \\
+                        y_1' = a\cos{x} + b\sin{x} - x(a\sin{x} - b\cos{x}) \\
+                        y_1'' = -2a\sin{x} + 2b\cos{x} - x(a\cos{x} + b\sin{x}) \\
+                    \end{array}
+                    \right.
+                \end{align*}
+
+                Dans $(E_7)$~:
+                \begin{align*}
+                    -2a\sin{x} + 2b\cos{x} - x(a\cos{x} + b\sin{x}) + x(a\cos{x} + b\sin{x}) = \cos{x} \\
+                    \iff
+                    -2a\sin{x} + 2b\cos{x} = \cos{x} \\
+                    \iff
+                    2b\cos{x} - 2a\sin{x} = \cos{x} \\
+                    \implies
+                    \left\{
+                    \begin{array}{l}
+                        2b = 1 \\
+                        -a = 0 \\
+                    \end{array}
+                    \right.
+                    \implies
+                    \left\{
+                    \begin{array}{l}
+                        a = 0 \\
+                        b = \frac{1}{2} \\\\
+                    \end{array}
+                    \right.
+                \end{align*}
+                \begin{align*}
+                    \implies
+                    y_1 = \frac{x\sin{x}}{2} \\
+                \end{align*}
+
+            \item Solution générale
+                \begin{equation*}
+                    y = y_0 + y_1 = \boxed{\lambda \cos{x} + \mu \sin{x} + \frac{x\sin{x}}{2}}
+                \end{equation*}
+
+        \end{enumerate}
+
     \paragraph{$(E_8)$}
         $y'' - 4y = (-4x + 3) e^{2x}$