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\begin{document}

{\bf Question}

Express the following system of first order differential equations
in matrix form, and find the eigenvalues and eigenvectors if the
associated coefficient matrix:

$$\frac{dx}{dt}=2x + 3y$$

$$\frac{dy}{dt} = 2x + y.$$

Hence find the general solution to the equations.

{\bf Answer} \vspace{.2in}

In matrix form: $\left( \begin{array}{c} \frac{dx}{dt} \\
\frac{dy}{dt} \end{array} \right) = \left( \begin{array}{cc} 2 &amp; 3
\\ 2 &amp; 1 \end{array} \right)\left( \begin{array}{c} x \\
y \end{array} \right)$
\begin{eqnarray*}
 \left| \begin{array}{cc} {2 - \lambda} &amp; {3} \\ {2} &amp; {1 - \lambda} \\
\end{array} \right| &amp; = &amp; (2 - \lambda)(1 - \lambda) - 6 \\ &amp; = &amp;
\lambda^2 - 3\lambda - 4 \\ &amp; = &amp; (\lambda -  4)(\lambda + 1) = 0
\\ {\rm so \ \ \ } \lambda &amp; = &amp; -1, 4
\end{eqnarray*}

$ \underline {\lambda = -1}$ \hspace{.2in} Solve $\left(
\begin{array}{rr} 3 &amp; 3\\ 2 &amp; 2\\ \end{array} \right)
\left( \begin{array}{r} x \\ y\\ \end{array} \right) = \left(
\begin{array}{r} 0 \\ 0 \\ \end{array} \right)$

or $\left. \begin{array}{rcl} {3x + 3y} &amp; = &amp; 0 \\ {2x + 2y} &amp; = &amp;
0 \\ \end{array} \right\} \begin{array}{lrcl} {\rm let\ \ } &amp; x &amp;
= &amp; \alpha \\ {\rm so\ \ } &amp; y &amp; = &amp; -\alpha \\ \end{array}$

Suitable eigenvector $\left( \begin{array}{c} \alpha \\ -\alpha
\\ \end{array} \right)$

$ \underline {\lambda = 4}$ \hspace{.2in} Solve $\left(
\begin{array}{rr} -2 &amp; {3}\\ {2} &amp; -3\\ \end{array} \right)
\left( \begin{array}{r} x \\ y\\ \end{array} \right) = \left(
\begin{array}{r} 0 \\ 0 \\ \end{array} \right)$

or $\left. \begin{array}{rcl} {-2x + 3y} &amp; = &amp; 0 \\ {2x - 3y} &amp; =
&amp; 0 \\ \end{array} \right\} \begin{array}{lrcl} {\rm let\ \ } &amp; x
&amp; = &amp; 3\beta \\ {\rm so\ \ } &amp; y &amp; = &amp; 2\beta \\ \end{array}$

Suitable eigenvector $\left( \begin{array}{c} 3\beta \\ 2\beta
\\ \end{array} \right)$

General solution to equations:
\begin{eqnarray*}
\left( \begin{array}{c} x \\ y \end{array} \right) &amp; = &amp; \left(
\begin{array}{c} \alpha \\ -\alpha \end{array} \right)e^{-t} +
\left( \begin{array}{c} 3\beta \\ 2\beta \end{array} \right)e^{4t}
\\ &amp; = &amp; \alpha \left( \begin{array}{c} 1 \\ -1 \end{array} \right)e^{-t}
+\beta \left( \begin{array}{c} 3 \\ 2 \end{array} \right)e^{4t}
\end{eqnarray*}


\end{document}
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