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 $𝐜\mathit{\psi }$ where $𝐜$ is the constant matrix and $\mathit{\psi }$ is the augmented set of independent solutions to the homogeneous equation, we try out a form like so:

$𝐗=𝐮\mathit{\psi }$

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

$𝐗=\mathit{\psi }(t){\mathit{\psi }}^{-1}(t_0){𝐗}^0+\mathit{\psi }(t)\underset{t_0}{\overset{t}{\int }}{\mathit{\psi }}^{-1}(s)𝐠(s)ds$

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