Linear Algebra with Differential Equations/Heterogeneous Linear Differential Equations/Variation of Parameters

As with the variation of parameters in the normal differential equations (a lot of similarities here!) we take a fundamental solution and by using a product with a to-be-found vector, see if we can come upon another independent solution by these means. In other words, since the general solution can be expressed as $\mathbf {c\psi }$ where $\mathbf {c}$ is the constant matrix and $\mathbf {\psi }$ is the augmented set of independent solutions to the homogeneous equation, we try out a form like so:

$\mathbf {X} =\mathbf {u\psi }$

And determine $\mathbf {u}$ to find a unique solution. The math is fairly straightforward and left as an exercise for the reader, and leaves us with:

$\mathbf {X} =\mathbf {\psi } (t)\mathbf {\psi } ^{-1}(t_{0})\mathbf {X} ^{0}+\mathbf {\psi } (t)\int _{t_{0}}^{t}\mathbf {\psi } ^{-1}(s)\mathbf {g} (s)ds$

... which is a fairly strong, striaghtforward, yet exceedingly complicated formula.