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authorAndreas Müller <andreas.mueller@ost.ch>2021-06-14 07:26:10 +0200
committerGitHub <noreply@github.com>2021-06-14 07:26:10 +0200
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+%
+% matrix-dgl.tex -- Matrix-Differentialgleichungen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{1.~Ordnung mit Skalaren}
+ \vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \begin{block}{Aufgabe}
+ Sei $a, x(t), x_0 \in \mathbb R$,
+ \[
+ \dot x(t) = ax(t),
+ \quad
+ x(0) = x_0
+ \]
+ \end{block}
+ \begin{block}{Potenzreihen-Ansatz}
+ Sei $a_k \in \mathbb R$,
+ \[
+ x(t) = a_0 + a_1t + a_2t^2 + a_3t^3 \ldots
+ \]
+ \end{block}
+ \end{column}
+ \begin{column}{0.48\textwidth}
+ \begin{block}{Lösung}
+ Einsetzen in DGL, Koeffizientenvergleich liefert
+ \[ x(t) = \exp(at) \, x_0, \]
+ wobei
+ \begin{align*}
+ \exp(at)
+ &= 1 + at + \frac{a^2t^2}{2} + \frac{a^3t^3}{3!} + \ldots \\
+ &{\color{gray}(= e^{at}.)}
+ \end{align*}
+ \end{block}
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{1.~Ordnung mit Matrizen}
+ \vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \begin{block}{Aufgabe}
+ Sei $A \in M_n$, $x(t), x_0 \in \mathbb R^n$,
+ \[
+ \dot x(t) = Ax(t),
+ \quad
+ x(0) = x_0
+ \]
+ \end{block}
+ \begin{block}{Potenzreihen-Ansatz}
+ Sei $A_k \in \mathbb M_n$,
+ \[
+ x(t) = A_0 + A_1t + A_2t^2 + A_3t^3 \ldots
+ \]
+ \end{block}
+ \end{column}
+ \begin{column}{0.48\textwidth}
+ \begin{block}{Lösung}
+ Einsetzen in DGL, Koeffizientenvergleich liefert
+ \[ x(t) = \exp(At) \, x_0, \]
+ wobei
+ \[
+ \exp(At)
+ = 1 + At + \frac{A^2t^2}{2} + \frac{A^3t^3}{3!} + \ldots
+ \]
+ \end{block}
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\egroup