Prove that a function satisfying a given equation must be the exponential function

by RoRi

November 16, 2015

If $f(x)$ is a function defined for all $x \in \mathbb{R}$ and such that

$f'(x) = cf(x) \qquad \text{for all } x$

prove that we must have

$f(x) = Ke^{cx}$

for some constant $K$ .

Proof. Let $g(x) = f(x)e^{-cx}$ . Then we can compute the derivative,

$g'(x) = -cf(x)e^{-cx} + f'(x) e^{-cx}.$

Since we know $f'(X) = cf(x)$ we then have

$g'(x) = -cf(x)e^{-cx} + cf(x)e^{-cx} = 0.$

Therefore, $g(x) = K$ for some constant $K$ and we have

$K = f(x)e^{-cx} \quad \implies \quad f(x) = Ke^{cx}. \qquad \blacksquare$

Stumbling Robot