# Thread: Proving These are Exponential

1. ## Proving These are Exponential

How would I go about proving that given

$f'(x)=f(x)-g(x)$
$g'(x)=g(x)-f(x)$

Then $f(x)$ and $g(x)$ must be exponential functions.

Thank you

2. Differentiate the first equation to get $f''(x) = f'(x) - g'(x) = 2f(x)$

And from there you should (probably) know how to solve it (since you're being asked this :P)

3. the problem here is that we don't know if $f$ is twice differentiable.

here's a way to proceed: we have $f'(x)-g'(x)=2\big(f(x)-g(x)\big),$ then $\big(f(x)-g(x)\big)'=2\big(f(x)-g(x)\big),$ so $f(x)-g(x)=c_1e^{2x},$ where $c_1\in\mathbb R.$

on the other hand is easy to see that $f(x)+g(x)=c_2$ for $c_2\in\mathbb R,$ whereat $f(x)=\dfrac{c_1e^{2x}+c_2}2$ and $g(x)=\dfrac{c_2-c_1e^{2x}}2,$ as required.