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flyingscorpio@arch-desktop 2021-09-20 08:39:41 +02:00
parent eb97264985
commit a9ca852ba7
1 changed files with 20 additions and 20 deletions
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analyse/exercices/main.tex
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@ -34,10 +34,10 @@
\right.
\end{align*}
\begin{align*}
y_1' - 2 y_1 = x^2
&\Leftrightarrow
\text{Dans } (E_1)
&\implies
2ax + b - 2(ax^2 + bx + c) = x^2 \\
&\Leftrightarrow
&\iff
\left\{
\begin{array}{l}
-2a = 1 \\
@ -45,7 +45,7 @@
b - 2c = 0 \\
\end{array}
\right.
\Leftrightarrow
\iff
\left\{
\begin{array}{l}
a = \sfrac{-1}{2} \\
@ -85,12 +85,12 @@
\right.
\end{align*}
\begin{align*}
3y_1' - 9y_1 = 7e^{3x}
&\Leftrightarrow 3(3ax + a)e^{3x} - 9axe^{3x} = 7e^{3x} \\
&\Leftrightarrow (9ax + 3a)e^{3x} - 9axe^{3x} = 7e^{3x} \\
&\Leftrightarrow 9ax + 3a - 9ax = 7 \\
&\Leftrightarrow 3a = 7 \\
&\Leftrightarrow a = \frac{7}{3}
\text{Dans } (E_2)
&\implies 3(3ax + a)e^{3x} - 9axe^{3x} = 7e^{3x} \\
&\iff (9ax + 3a)e^{3x} - 9axe^{3x} = 7e^{3x} \\
&\iff 9ax + 3a - 9ax = 7 \\
&\iff 3a = 7 \\
&\iff a = \frac{7}{3}
\end{align*}
\begin{align*}
\implies y_1 = \frac{7}{3}xe^{3x}
@ -123,10 +123,10 @@
\right.
\end{align*}
\begin{align*}
2y_1' - 4y_1 = 5e^{3x}
&\Leftrightarrow 6ae^{3x} - 4ae^{3x} = 5e^{3x} \\
&\Leftrightarrow 2a = 5 \\
&\Leftrightarrow a = \frac{5}{2}
\text{Dans } (E_3)
&\iff 6ae^{3x} - 4ae^{3x} = 5e^{3x} \\
&\iff 2a = 5
\iff a = \frac{5}{2}
\end{align*}
\begin{align*}
\implies y_1 = \frac{5}{2}e^{3x}
@ -159,30 +159,30 @@
\right.
\end{align*}
\begin{align*}
y_1' + 4y_1 = 3\cos(2x)
&\Leftrightarrow -2a\sin(2x) + 2b\cos(2x) + 4(a\cos(2x) + b\sin(2x)) = 3\cos(2x) \\
&\Leftrightarrow
\text{Dans } (E_4)
&\implies -2a\sin(2x) + 2b\cos(2x) + 4(a\cos(2x) + b\sin(2x)) = 3\cos(2x) \\
&\iff
\left\{
\begin{array}{l}
4a + 2b = 3 \\
4b - 2a = 0 \\
\end{array}
\right.
\Leftrightarrow
\iff
\left\{
\begin{array}{l}
4a + 2b = 3 \\
4a - 8b = 0 \\
\end{array}
\right.
\Leftrightarrow
\iff
\left\{
\begin{array}{l}
10b = 3 \\
4b - 2a = 0 \\
\end{array}
\right. \\
&\Leftrightarrow
&\iff
\left\{
\begin{array}{l}
b = \sfrac{3}{10} \\