A Neat Argument For the Uniqueness of $e^x$

In my advanced Calculus 1 class I teach a quick unit on differential equations. We don’t have the tools to solve them since we haven’t done integrals, but I talk about what differential equations are and how you can check whether you have a solution.

And then I spend a day in lab discussing exponential growth, and how the differential equation $y’ = ry$ implies that $y = Ce^{rt}$ for some constants $C$ and $r$. I’ve been telling my students that while it’s easy to check that this is a solution, we don’t have the tools to prove it’s the only family of solutions.

But today thanks to reddit, I discovered that that isn’t quite true. You can prove that $Ce^x$ is the only solution to this differential equation with a simple argument.

Suppose $f(x)$ is a function that satisfies $y’ = r y$, that is, suppose $f’(x) = r f(x)$. Then consider the derivative of $f(x) e^{-rx}$. By the product rule, we have

\[ \begin{aligned} \frac{d}{dx} f(x) e^{-rx} &= f’(x) e^{-rx} + f(x) (-r)e^{-rx} \\ &= r f(x) e^{-rx} - rf(x) e^{-rx} = 0. \end{aligned} \]

Thus we see that $f(x)/e^{rx}$ must be a constant; and thus $f(x) = C e^{rx}$. So this family of solutions is unique.