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

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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.