Solve the Linear Dynamical System $\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} =A\mathbf{x}$ by Diagonalization
Problem 667
(a) Find all solutions of the linear dynamical system
\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} =\begin{bmatrix}
1 & 0\\
0& 3
\end{bmatrix}\mathbf{x},\]
where $\mathbf{x}(t)=\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}$ is a function of the variable $t$.
(b) Solve the linear dynamical system
\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}=\begin{bmatrix}
2 & -1\\
-1& 2
\end{bmatrix}\mathbf{x}\]
with the initial value $\mathbf{x}(0)=\begin{bmatrix}
1 \\
3
\end{bmatrix}$.
