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Add E7 equa diff 2nd ordre

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flyingscorpio@pinebookpro 2021-09-21 15:54:45 +02:00
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analyse/exercices/main.tex
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@ -349,12 +349,96 @@
y = y_0 + y_1 = \boxed{\lambda e^{-x} + \mu e^{x} - x^3 - 6x} y = y_0 + y_1 = \boxed{\lambda e^{-x} + \mu e^{x} - x^3 - 6x}
\end{equation*} \end{equation*}
\end{enumerate} \end{enumerate}
\paragraph{$(E_7)$} \paragraph{$(E_7)$}
$y'' + y = \cos{x}$ $y'' + y = \cos{x}$
\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{$(E_8)$} \paragraph{$(E_8)$}
$y'' - 4y = (-4x + 3) e^{2x}$ $y'' - 4y = (-4x + 3) e^{2x}$