Integrating Factors

It's generally not easy to recognize how to integrate even some simple first order, linear D.E.'s (F.O.L.'s from now on)

Sometimes, we can magically multiply by something called an integrating factor to find a solution.

Given a F.O.L.:

y' + p(t)y = g(t)

Define

$$\mu(t) = e^{\int{p(t)}}$$

Now, multiply the F.O.L. by $\mu(t)$

$$\mu(t)y' + \mu(t) p(t) y = \mu(t) g(t)$$

Believe it or not, life just got simpler. The left hand side of the equation is the derivative of $\mu(t) y$.