Variation of parameters

We can use variation of parameters to solve a first order inhomogeneous ODE of the form

\dot y + p(t) y = q(t).

We first solve the homogeneous ODE $\dot y_h + p(t) y_h = 0$ . Then set $y = u(t) y_h$ and plug into the inhomogeneous ODE. Terms will cancel.

Variation of parameters for an ODE system

We can apply a similar technique to an ODE system $\dot{\mathbf x} = A \mathbf x + \mathbf q$ . First write the homogeneous solution in terms of the fundamental matrix as $\mathbf x = e^{At} \mathbf C$ , then replace $\mathbf C$ with a vector-valued function $\mathbf u(t)$ . Plugging into the ODE, we find

\begin{align*} Ae^{At} \mathbf u(t) + e^{At} \dot{\mathbf u}(t) &= A\mathbf x + \mathbf q(t) \\ A\mathbf x + e^{At} \dot{\mathbf u}(t) &= A\mathbf x + \mathbf q(t) \\ e^{At} \dot{\mathbf u}(t) &= \mathbf q(t) \\ \mathbf u(t) &= \int e^{-At} \mathbf q(t) \, \mathrm dt. \end{align*}