Consider $v'=\begin{bmatrix} 0&1 \\ -2 & 3 \end{bmatrix}v+\begin{bmatrix} 1\\ 1 \end{bmatrix}$ with transition matrix of associated homogeneous system $\Phi(x,0)=\begin{bmatrix} 2\mathit{e}^{x}-\mathit{e}^{2x} &-\mathit{e}^{x}+\mathit{e}^{2x} \\ 2\mathit{e}^{x}-2\mathit{e}^{2x} &-\mathit{e}^{x}+2\mathit{e}^{2x} \end{bmatrix}$. Find a general solution to the linear nonhomogeneous system with $v(0)=0$.
I already verified that $\Phi(x,0)$ is the transition matrix for the associated homogeneous system. Do I use the formula $\Phi(x,0)\mathbf{\xi}+\int_{\tau}^{t}\Phi(t,\eta)g(\eta)d\eta$ to the general solution to the linear nonhomogeneous system, where $\xi=0$, $\tau=0$, and $g(t)=\begin{bmatrix} 1\\ 1 \end{bmatrix}$?
If I use that formula, how do I integrate the matrix? Any hints and ideas are greatly appreciated!
I get $\begin{bmatrix} 0 &0 \\ 0&0 \end{bmatrix}+\int_{0}^{t}\Phi(t,\eta)\begin{bmatrix} 1\\ 1 \end{bmatrix}d\eta$, but I am not sure this is correct.