QuestionFind the general solution of the system $$\mathbf{x}^{\prime}=\left(\begin{array}{rr} -2 & 1 \\ 1 & -2 \end{array}\right) \mathbf{x}+\binom{2 e^{-t}}{3 t}=\mathbf{A} \mathbf{x}+\mathbf{g}(t)$$

Studdy Solution
Combine the homogeneous and particular solutions to form the general solution:
$\mathbf{x}(t) = \mathbf{x}_h(t) + \mathbf{x}_p(t)$
$\mathbf{x}(t) = c_1 e^{-t} \begin{pmatrix} 1 \\ 1 \end{pmatrix} + c_2 e^{-3t} \begin{pmatrix} 1 \\ -1 \end{pmatrix} + \begin{pmatrix} 2 e^{-t} \\ 3 t \end{pmatrix}$
The general solution of the system is:
$\mathbf{x}(t) = c_1 e^{-t} \begin{pmatrix} 1 \\ 1 \end{pmatrix} + c_2 e^{-3t} \begin{pmatrix} 1 \\ -1 \end{pmatrix} + \begin{pmatrix} 2 e^{-t} \\ 3 t \end{pmatrix}$