A calculus of the absurd

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

\[\frac {dy}{dx} + p(x)y = q(x)\]

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 \(y e^{f(x)}\) is

\begin{equation} \frac {dy}{dx} e^{f} + e^{f}\frac {df}{dx}y \end{equation}

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

\begin{equation} \frac {dy}{dx} e ^ {f(x)} + p(x) e^{f(x)} y = q(x) e ^{f(x)} \end{equation}

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 \(p(x)\). ⁸⁸⁸⁸ 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

\begin{equation} \frac {d}{dx} \left [ ye^{f(x)} \right ] = q(x) e ^{f(x)} \end{equation}

And thus we can solve the equation by integrating.