%
% hyperbolisch.tex
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\bgroup
\definecolor{darkgreen}{rgb}{0,0.6,0}
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Hyperbolische Funktionen}
\vspace{-15pt}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.48\textwidth}
\begin{block}{Differentialgleichung}
\vspace{-10pt}
\begin{align*}
\ddot{y} &= y
\;\Rightarrow\;
\frac{d}{dt}
\begin{pmatrix}y\\y_1\end{pmatrix}
=
\begin{pmatrix}0&1\\1&0\end{pmatrix}
\begin{pmatrix}y\\y_1\end{pmatrix}
\\
y(0)&=a,\qquad y'(0)=b
\end{align*}
\end{block}
\vspace{-10pt}
\uncover<2->{%
\begin{block}{Lösung}
\vspace{-13pt}
\begin{align*}
\lambda^2-1&=0
\uncover<3->{
\qquad\Rightarrow\qquad \lambda=\pm 1
}
\\
\uncover<4->{
y(t)&=Ae^t+Be^{-t}}
\uncover<5->{
\Rightarrow
\left\{
\arraycolsep=1.4pt
\begin{array}{rcrcr}
A&+&B&=&a\\
A&-&B&=&b
\end{array}
\right.}
\\
&\uncover<6->{
=\frac{a+b}2e^t + \frac{a-b}2e^{-t}}
\\
&\uncover<7->{=
a{\color{darkgreen}\frac{e^t+e^{-t}}2} + b{\color{red}\frac{e^t-e^{-t}}2}}
\end{align*}
\end{block}}
\end{column}
\begin{column}{0.49\textwidth}
\uncover<8->{%
\begin{block}{Potenzreihe}
\vspace{-12pt}
\begin{align*}
K&=\begin{pmatrix}0&1\\1&0\end{pmatrix}
\uncover<10->{\quad\Rightarrow\quad K^2=I}
\\
\uncover<9->{
e^{Kt}
&=
I+K+\frac1{2!}K^2 + \frac{1}{3!}K^3 + \frac{1}{4!}K^4+\dots
}
\\
\uncover<11->{
&=
\biggl( 1+\frac{t^2}{2!} + \frac{t^4}{4!}+\dots \biggr)I
}
\\
\uncover<11->{
&\qquad
+\biggl(t+\frac{t^3}{3!}+\frac{t^5}{5!}+\dots\biggr)K
}
\\
\uncover<12->{
&=
I{\,\color{darkgreen}\cosh t}  + K{\,\color{red}\sinh t}
}
\\
\uncover<13->{
\begin{pmatrix}y(t)\\y_1(t)\end{pmatrix}
&=
e^{Kt}\begin{pmatrix}a\\b\end{pmatrix}
}
\uncover<14->{
=
\begin{pmatrix}
a{\,\color{darkgreen}\cosh t} + b{\,\color{red}\sinh t}\\
a{\,\color{red}\sinh t} + b{\,\color{darkgreen}\cosh t}
\end{pmatrix}
}
\end{align*}
\end{block}}
\end{column}
\end{columns}
\end{frame}
\egroup