# Method of undetermined coefficients for non-homogeneous linear system with two constant vectors

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:

$$g_1=-A^{-1}b_0 \\ g_2=-A^{-1}b_1$$

• Why is this getting downvoted? – Steve Apr 12 '16 at 22:32

$$A(g_1+g_2t)+b_0+b_1t$$
$$=Ag_1+Ag_2t+b_0+b_1t$$
$$=Ag_1+b_0+(Ag_2+b_1)t$$
$g_1$ and $g_2$ can now be determined by solving the above system of equations.