13.2 Integrating Factors

$e^{x}$ shows up a lot in differential equations, because it has properties that are helpful when we differentiate it. One way in which it helps us is in solving first-order linear differential equations, which are equations of the form

This can be solved using the product rule. If we define a function $f(x)$, we can write by the product rule that the derivative of $ye^{f(x)}$ is

This doesn’t immediately look like our equation, but if we multiply through by $e^{f}$, we get that

What we can do here is write that the left hand side is equal to the derivative of $ye^{f(x)}$. This only works, however, if the derivative of $f(x)$ is equal to $p(x)$. ¹¹ 1 This is because $\begin{aligned} \frac{d}{dx}\left[ye^{f(x)}\right]&=\frac{d}{dx}{y}e^{f(x)}+y\frac{d}{dx}\left[e^{f(x)}\right]\\ &=\frac{dy}{dx}e^{f(x)}+y\frac{d}{dx}\left[f(x)\right]e^{f(x)}\\ \end{aligned}$ And if $\begin{aligned} f(x)=\int p(x)dx\end{aligned}$ then $\begin{aligned} \frac{d}{dx}\left[f(x)\right]=p(x)\end{aligned}$ And thus $\frac{d}{dx}\left[ye^{f(x)}\right]=\frac{dy}{dx}e^{f(x)}+p(x)e^{f(x)}y$ which is just the left-hand side of the equation. If it is, we can write that

And thus we can solve the equation by integrating.