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 $x_1=0, x_2=1, x_3=5$.
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