## 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}$.