<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\newcommand{\ud}{\underline}
\newcommand{\lin}{\int_0^\infty}
\parindent=0pt
\begin{document}


{\bf Question}

Obtain a fundamental matrix $\Phi(t)$ for each of the systems which
satisfies the initial conditions $\Phi(0)=I$, and
hence solve the above problems with the initial conditions $$ {\bf
x}(0)=\pmatrix{1 \cr 1\cr} $$
\begin{description}
\item[(a)] $\ds \pmatrix{3 &amp;-2\cr 2 &amp;-2\cr}$
\item[(b)]$\ds \pmatrix{4 &amp;-3\cr 8 &amp;-6\cr}$
\item[(c)] $\ds \pmatrix{2 &amp;-1\cr 3 &amp;-2\cr}$
\item[(d)] $\ds \pmatrix{1 &amp;1\cr 4 &amp;-2\cr}$
\item[(a)] $\ds \pmatrix{2 &amp;-5\cr 1 &amp;-2\cr}$
\item[(e)] $\ds \pmatrix{-1 &amp;-4\cr 1 &amp;-1\cr}$
\item[(f)] $\ds \pmatrix{5 &amp;-1\cr 3 &amp;1\cr}$
\item[(g)] $\ds \pmatrix{3 &amp;-4\cr 1 &amp;-1\cr}$
\end{description}


\vspace{.5in}

{\bf Answer}

\begin{description}
\item[(a)] $$ \Phi(t)=\pmatrix{-1/3e^{-t}+4/3e^{2t} &amp;
2/3e^{-t}-2/3e^{2t} \cr
                 -2/3e^{-t}+2/3e^{2t} &amp; 4/3e^{-t}-1/3e^{2t} \cr}
$$

\item[(b)]
$$
\Phi(t)=\pmatrix{3-2e^{-2t}&amp;-3/2+3/2e^{-2t} \cr
                 4-4e^{-2t}&amp;  -2+3e^{-2t} \cr}
$$

\item[(c)]
$$
\Phi(t)=\pmatrix{3/2e^{t}-1/2e^{-t} &amp; -1/2e^{t}+1/2e^{-t} \cr
                 3/2e^{t}-3/2e^{-t} &amp; -1/2e^{t}+3/2e^{-t} \cr}
$$

\item[(d)]
$$
\Phi(t)=\pmatrix{1/5e^{-3t}+4/5e^{2t}&amp;-1/5e^{-3t}+1/5e^{2t} \cr
                -4/5e^{-3t}+4/5e^{2t}&amp; 4/5e^{-3t}+1/5e^{2t} \cr}
$$

\item[(e)]
$$
\Phi(t)=\pmatrix{\cos t +2\sin t&amp; -5\sin t \cr
                 \sin t         &amp; \cos t -2\sin t \cr}
$$


\item[(f)]
$$
\Phi(t)=\pmatrix{e^{-t}\cos 2t &amp; -2e^{-t}\sin 2t \cr
                 1/2e^{-t}\sin 2t &amp; e^{-t}\cos 2t \cr}
$$


\item[(g)]
$$
\Phi(t)=\pmatrix{-1/2e^{2t}+3/2e^{4t}&amp;1/2e^{2t}-1/2e^{4t}\cr
                 -3/2e^{2t}+3/2e^{4t}&amp;3/2e^{2t}-1/2e^{4t}\cr}
$$

\item[(h)]
$$
\Phi(t)=\pmatrix{2te^t+e^t&amp;-4te^t \cr
                 te^t     &amp;e^t-2te^t\cr}
$$
\end{description}


\end{document}
</pre></body></html>