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Introduction
Not every differential equation will politely separate its variables for us. For first-order linear equations, there’s another powerful tool: the
integrating factor method . This method works by multiplying the entire equation by a carefully chosen function — an “integrating factor” — that transforms the left side into a single derivative.
Here’s the intuition: Every equation of the form
\begin{equation*}
\frac{dy}{dx} + P(x)y = Q(x)
\end{equation*}
can be multiplied by a special factor to give you
\begin{equation*}
\boxed{\phantom{M}}\cdot \frac{dy}{dx} + \ \boxed{\phantom{M}}\cdot P(x)y = \ \boxed{\phantom{M}}\cdot Q(x)
\end{equation*}
and the whole left side becomes a single derivative:
\begin{equation*}
\frac{d}{dx}\big[\ \boxed{\phantom{M}}\cdot y\big] =\ \boxed{\phantom{M}}\cdot Q(x).
\end{equation*}
The goal of this chapter is to learn where this mysterious factor comes from, how to find it, and how to use it to solve any first-order linear differential equation.
