13.2 Integrating Factors

exe^{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)f(x), we can write by the product rule that the derivative of yef(x)ye^{f(x)} is

dydxef+efdfdxy\frac{dy}{dx}e^{f}+e^{f}\frac{df}{dx}y (13.1)

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

dydxef(x)+p(x)ef(x)y=q(x)ef(x)\frac{dy}{dx}e^{f(x)}+p(x)e^{f(x)}y=q(x)e^{f(x)} (13.2)

What we can do here is write that the left hand side is equal to the derivative of yef(x)ye^{f(x)}. This only works, however, if the derivative of f(x)f(x) is equal to p(x)p(x). 11 1 This is because ddx[yef(x)]=ddxyef(x)+yddx[ef(x)]=dydxef(x)+yddx[f(x)]ef(x)\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 f(x)=p(x)𝑑x\begin{aligned} f(x)=\int p(x)dx\end{aligned} then ddx[f(x)]=p(x)\begin{aligned} \frac{d}{dx}\left[f(x)\right]=p(x)\end{aligned} And thus ddx[yef(x)]=dydxef(x)+p(x)ef(x)y\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

ddx[yef(x)]=q(x)ef(x)\frac{d}{dx}\left[ye^{f(x)}\right]=q(x)e^{f(x)} (13.3)

And thus we can solve the equation by integrating.