How would I go about proving that given
$\displaystyle f'(x)=f(x)-g(x)$
$\displaystyle g'(x)=g(x)-f(x)$
Then $\displaystyle f(x)$ and $\displaystyle g(x)$ must be exponential functions.
Thank you
the problem here is that we don't know if $\displaystyle f$ is twice differentiable.
here's a way to proceed: we have $\displaystyle f'(x)-g'(x)=2\big(f(x)-g(x)\big),$ then $\displaystyle \big(f(x)-g(x)\big)'=2\big(f(x)-g(x)\big),$ so $\displaystyle f(x)-g(x)=c_1e^{2x},$ where $\displaystyle c_1\in\mathbb R.$
on the other hand is easy to see that $\displaystyle f(x)+g(x)=c_2$ for $\displaystyle c_2\in\mathbb R,$ whereat $\displaystyle f(x)=\dfrac{c_1e^{2x}+c_2}2$ and $\displaystyle g(x)=\dfrac{c_2-c_1e^{2x}}2,$ as required.