# 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:
$\mathbf{X} = \mathbf{\psi}(t) \mathbf{\psi}^{-1}(t_0) \mathbf{X}^0 + \mathbf{\psi}(t) \int_{t_0}^t \mathbf{\psi}^{-1}(s) \mathbf{g}(s) ds$