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Generalize discriminant for equa diff 2nd ordre

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flyingscorpio@arch-desktop 2021-09-20 08:27:39 +02:00
parent e0335b2b42
commit eb97264985
1 changed files with 40 additions and 3 deletions
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43
analyse/main.tex
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@ -408,7 +408,24 @@
\color{red}{y_0 = \lambda e^{r_1 x} + \mu e^{r_2 x}} \quad \text{ où } \lambda, \mu \in \mathbb{R}
\end{equation*}
\item si $\Delta = 0$, l'équation caractéristique possède une racine double $r_0 = \frac{-b}{2a}$
\item si $\Delta = 0$, l'équation caractéristique possède une racine double $r_0$~:
\begin{align*}
\left\{
\begin{array}{l}
r_0 = \frac{-b - \sqrt{\Delta}}{2a} \\\\
r_0 = \frac{-b + \sqrt{\Delta}}{2a} \\
\end{array}
\right.
\implies
\left\{
\begin{array}{l}
r_0 = \frac{-b - 0}{2a} \\\\
r_0 = \frac{-b + 0}{2a} \\
\end{array}
\right.
\implies
r_0 = \frac{-b}{2a}
\end{align*}
Les solutions de $(E_0)$ sont alors~:
@ -423,8 +440,28 @@
r_2 = \alpha - i\beta \\
\end{array}
\right.$
On trouve $\alpha$ et $\beta$ avec la même formule que pour $\Delta > 0$, en prenant $\sqrt{\Delta} = i \sqrt{|\Delta|}$.
\begin{align*}
\left\{
\begin{array}{l}
r_1 = \frac{-b - \sqrt{\Delta}}{2a} \\\\
r_2 = \frac{-b + \sqrt{\Delta}}{2a} \\
\end{array}
\right.
\implies
\left\{
\begin{array}{l}
r_1 = \frac{-b - i\sqrt{|\Delta|}}{2a} \\\\
r_2 = \frac{-b + i\sqrt{|\Delta|}}{2a} \\
\end{array}
\right.
\implies
\left\{
\begin{array}{l}
\alpha = \frac{-b}{2a} \\\\
\beta = \frac{i\sqrt{|\Delta|}}{2a}\\
\end{array}
\right.
\end{align*}
Les solutions de $(E_0)$ sont alors~: