From ecc99689218d54f550a3e393c6536592d3678e93 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Tue, 9 Mar 2021 11:10:02 +0100
Subject: potenzreihenmethode

---
 vorlesungen/slides/5/potenzreihenmethode.tex | 93 ++++++++++++++++++++++++++++
 1 file changed, 93 insertions(+)
 create mode 100644 vorlesungen/slides/5/potenzreihenmethode.tex

(limited to 'vorlesungen/slides/5/potenzreihenmethode.tex')

diff --git a/vorlesungen/slides/5/potenzreihenmethode.tex b/vorlesungen/slides/5/potenzreihenmethode.tex
new file mode 100644
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--- /dev/null
+++ b/vorlesungen/slides/5/potenzreihenmethode.tex
@@ -0,0 +1,93 @@
+%
+% potenzreihenmethode.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Potenzreihenmethode}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Lineare Differentialgleichung}
+\vspace{-12pt}
+\begin{align*}
+y'&=ay&&\Rightarrow&y'-ay&=0
+\\
+y(0)&=C
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Potenzreihenansatz}
+\vspace{-12pt}
+\begin{align*}
+y(x)
+&=
+a_0+ a_1x + a_2x^2 + \dots
+\\
+y(0)&=a_0=C
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<3->{%
+\begin{block}{Lösung}
+\vspace{-12pt}
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcrcr}
+\uncover<3->{ y'(x)}
+	\uncover<5->{
+	&=&\phantom{(} a_1\phantom{\mathstrut-aa_0)}
+	&+& 2a_2\phantom{\mathstrut-aa_1)}x
+		&+& 3a_3\phantom{\mathstrut-aa_2)}x^2
+			&+& 4a_4\phantom{\mathstrut-aa_3)}x^3
+				&+& \dots}\\
+\uncover<3->{-ay(x)}
+	\uncover<6->{
+	&=&\mathstrut-aa_0 \phantom{)}
+	&-&  aa_1\phantom{)}x
+		&-&  aa_2\phantom{)}x^2
+			&-&  aa_3\phantom{)}x^3
+				&-& \dots}\\[2pt]
+\hline
+\\[-10pt]
+\uncover<3->{0}
+	\uncover<7->{
+	&=&(a_1-aa_0)
+	&+& (2a_2-aa_1)x
+		&+& (3a_3-aa_2)x^2
+			&+& (4a_4-aa_3)x^3
+				&+& \dots}\\
+\end{array}
+\]
+\begin{align*}
+\uncover<4->{
+a_0&=C}\uncover<8->{,
+\quad
+a_1=aa_0=aC}\uncover<9->{,
+\quad
+a_2=\frac12a^2C}\uncover<10->{,
+\quad
+a_3=\frac16a^3C}\uncover<11->{,
+\dots
+a_k=\frac1{k!}a^kC}
+\hspace{3cm}
+\\
+\uncover<4->{
+\Rightarrow y(x) &= C}\uncover<8->{+Cax}\uncover<9->{ + C\frac12(ax)^2}
+\uncover<10->{ + C \frac16(ac)^3}
+\uncover<11->{ + \dots+C\frac{1}{k!}(ax)^k+\dots}
+\ifthenelse{\boolean{presentation}}{
+\only<12>{
+=
+C\sum_{k=0}^\infty \frac{(ax)^k}{k!}}
+}{}
+\uncover<13->{=
+Ce^{ax}}
+\end{align*}
+\end{block}}
+\end{frame}
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