Integrating Factors

Integrating Factors

A first order equation of the form

can always be turned into a separable equation. We do this by introducing what's known as an integrating factor. What we're looking for is a function that we can multiply both sides of the equation by and that will turn the left side into the derivative of a single function, like this

By applying the product rule to the right side of this we get

Then by matching up the terms we can see that

That means that the function we're looking for is

Here is an example of how it works.

Solve the differential equation

First, the integrating factor we want is

So we get

We have solved for

. At this point, there is often nothing much else you can do. The family of functions on the right may be difficult to integrate and manipulate into a simpler form. In the case at hand, we can actually integrate by parts to get something simpler. Just consider the integral part:

Putting this all together we find that