Chapter 5 Integrating Factor
This chapter introduces the next solution method called the Integrating Factor Method (IF method). This important technique solves any first-order linear differential equation using a reversed version of the product rule followed by direct integration.
Recall, the Direct Integration method provides a straightforward way to solve differential equations when the equation can be written in the form:
\begin{equation*}
\frac{d}{dx}[g(x,y)] = f(x) \text{.}
\end{equation*}
In such cases, solving the equation is as simple as integrating both sides with respect to \(x\text{.}\) As you will see in the discussion that follow, the integrating factor method is essentially a direct integration problem in disguise. The difference here is that you must perform an initial step to rewrite the differential equation into a form for which direct integration applies.
The key idea is to multiply the differential equation by an integrating factor that simplifies it into a form that can be solved by direct integration. In the next sections, we will explore how this is done, step by step.
