Solving $f'(x) = -g(x)/x$, $g'(x) = f(x)/x$

Find all differentiable functions $f$ and $g$ at $(0,∞)$ such that $f'(x) = -g(x)/x$ and $g'(x) = f(x)/x$ for all $x > 0$.

I tried to determine the second derivative of $f$, then substituted $g(x)$ and $g'(x)$ to $f''(x)$. In the end, I obtained the Cauchy-Euler equation. Is this method correct? Is there any other way to answer this question?

Could you please help me? How can I solve this problem? Thank you.