## Solving a System of Differential Equation by Finding Eigenvalues and Eigenvectors

## Problem 668

Consider the system of differential equations

\begin{align*}

\frac{\mathrm{d} x_1(t)}{\mathrm{d}t} & = 2 x_1(t) -x_2(t) -x_3(t)\\

\frac{\mathrm{d}x_2(t)}{\mathrm{d}t} & = -x_1(t)+2x_2(t) -x_3(t)\\

\frac{\mathrm{d}x_3(t)}{\mathrm{d}t} & = -x_1(t) -x_2(t) +2x_3(t)

\end{align*}

**(a)** Express the system in the matrix form.

**(b)** Find the general solution of the system.

**(c)** Find the solution of the system with the initial value $x_1=0, x_2=1, x_3=5$.