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 where is the constant matrix and is the augmented set of independent solutions to the homogeneous equation, we try out a form like so:
And determine to find a unique solution. The math is fairly straightforward and left as an exercise for the reader, and leaves us with:
... which is a fairly strong, striaghtforward, yet exceedingly complicated formula.