# Solving Differential Equations Using an Integrating Factor

A clever method for solving differential equations (DEs) is in the form of a linear first-order equation. This method involves multiplying the entire equation by an *integrating factor.* A linear first-order equation takes the following form:

To use this method, follow these steps:

Calculate the integrating factor.

Multiply the DE by this integrating factor.

Restate the left side of the equation as a single derivative.

Integrate both sides of the equation and solve for

*y**.*

To help you understand how multiplying by an integrating factor works, the following equation is set up to practically solve itself — that is, if you know what to do:

Notice that this is a linear first-degree DE, with

and *b*(*x*) = 0. You now tweak this equation by multiplying every term by *x*^{2} (you see why shortly):

Next, you use algebra to do a little simplifying and rearranging:

Here’s where you appear to get extremely lucky: The two terms on the left side of the equation just happen to be the *result* of the application of the Product Rule to the expression *y *· *x*^{2}:

Notice that the right side of this equation is exactly the same as the left side of the previous equation. So you can make the following substitution:

Now, to undo the derivative on the left side, you integrate both sides, and then you solve for *y**:*

To check this solution, you plug this value of *y* back into the original equation:

The previous example works because you found a way to multiply the entire equation by a factor that made the left side of the equation look like a derivative resulting from the Product Rule. Although this looked lucky, if you know what to multiply by, *every* linear first-order DE can be transformed in this way. Recall that the form of a linear first-order DE is as follows:

The trick is to multiply the DE by an *integrating factor* based on *a*(*x*). Here’s the integrating factor:

For example, in the previous problem, you know that

So here’s how to find the integrating factor:

Remember that 2 ln *x* = ln *x*^{2}, so:

As you can see, the integrating factor *x*^{2}* *is the exact value that you multiplied by to solve the problem. To see how this process works now that you know the trick, here’s another DE to solve:

In this case, *a*(*x*) = 3, so compute the integrating factor as follows:

Now multiply every term in the equation by this factor:

If you like, use algebra to simplify the right side and rearrange the left side:

Now you can see how the left side of this equation looks like the *result* of the Product Rule applied to evaluate the following derivative:

Because the right side of this equation is the same as the left side of the previous equation, you can make the following substitution:

Notice that you change the left side of the equation using the Product Rule *in reverse.* That is, you’re expressing the whole left side as a single derivative. Now you can integrate both sides to undo this derivative:

Now solve for *y *and simplify:

To check this answer, substitute this value of *y* back into the original DE:

As if by magic, this answer checks out, so the solution is valid.