Find the solution to the following initial value problem $$x'=Ax+g(t), \quad x(0)=\begin{bmatrix} -2\\ 1\\ 4\end{bmatrix},\quad A=\begin{bmatrix} 6 & 3 & -2\\ -4 & -1 & 2\\ 13 & 9 & -3\end{bmatrix},\quad g(t)=\begin{bmatrix} 1\\ t\\ \sin t\end{bmatrix} $$
I am not sure how to solve this problem. I know how to get the general solution of form $x(t)=c_{1}x_{1}(t)+c_{2}x_{2}(t)+c_{3}x_{3}(t)$, and know I need to look for a solution of form $x(t)=ta+b+c\sin t+d\cos t$, but unsure how to do so.