Suppose I have a system of non-homogeneous linear first order differential equations:
$$ x'=A x+b_0+b_1t $$
Where $A$ is a $2\times2$ invertible matrix, $b_0$ and $b_1$ are:
$$ b_0 = \begin{pmatrix} r\\ u\\ \end{pmatrix}, \qquad b_1 = \begin{pmatrix} m\\ n\\ \end{pmatrix} $$
A particular solution for this system is given:
$$ x_p=g_1+g_2t $$
where $g_1$ and $g_2$ are constant vectors. Would I be able to find these constant vectors by determining them independently? Meaning I split them up like this:
$$ g_1=-A^{-1}b_0 \\ g_2=-A^{-1}b_1 $$
Never mind, I figured it out.
$$ A(g_1+g_2t)+b_0+b_1t $$
$$ =Ag_1+Ag_2t+b_0+b_1t $$
$$ =Ag_1+b_0+(Ag_2+b_1)t $$
$g_1$ and $g_2$ can now be determined by solving the above system of equations.