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.