Integrating factor

For first order linear ODEs the integrating factor method can be used to find a particular solution without first finding the homogeneous solution. Consider

\dot x + p(t) x = f(t).

We first take the antiderivative $P(t) = \int p(t) dx$ . Multiply both sides of the above equation by $e^{P(t)}$ to get

e^{P(t)} (\dot x + p(t) x) = e^{P(t)} f(t).

Then notice that $e^{P(t)}(\dot x + p(t) x) = \frac{d}{dt} \left[ e^{P(t)} x \right]$ .

\begin{align*} \frac{d}{dt} \left[ e^{P(t)} x \right] &= e^{P(t)} f(t) \\ e^{P(t)} x &= \int e^{P(t)} f(t) dt \\ x &= e^{-P(t)} \int e^{P(t)} f(t) dt. \end{align*}

The benefit of this method is that we can find a particular solution immediately, unlike variation of parameters; however the ODE must be first order and $e^{P(t)} f(t)$ must be easy to integrate.