The integrating factor we will look at on this page applies to first-order, linear differential equations of the form \(\displaystyle{ \frac{dy}{dt} + p(t)y = g(t) }\).
The idea of the technique of integrating factors is deceptively simple, yet quite powerful. When you have a first-order, linear differential equation of the form
\(\displaystyle{ \frac{dy}{dt} + p(t)y = g(t) }\)
and you multiply this equation by the generated integrating factor
\( \mu(t) = \exp \int{ p(t)~dt }\) [ what does exp mean? ]

This converts the differential equation into the form
\(\displaystyle{ \frac{d}{dt}[ \mu(t)y] = \mu(t)g(t) }\)

You can then integrate to get \(\displaystyle{ \mu(t)y }\) and divide by \(\mu(t)\) to solve for \(y\).
(Note: We will not go through the derivation of the integrating factor here at this time. However, going through the derivation in your textbook will really help you understand what is going on here.)
Okay, let's watch some videos, so we can see how this works.

Here is a good introduction to integrating factors that is not too long.

Here are a couple of complete examples using integrating factors to solve first-order, linear differential equations. In the first video, he works an example and explains integrating factors. In the second video, he expands on his discussion, giving more detail, while working a second example. In both videos, he explains how the product rule from calculus relates to this technique.

Okay, time for some practice problems.