What functions $u(x)$ and $v(x)$ satisfy $u(x) = v'(x)$ and $u'(x) = v(x)$?

Question

I recently came across a problem in a calculus textbook which involved functions satisfying the relation $u(x) = v'(x)$ and $u'(x) = v(x)$. The problem didn't list any specific functions that for which this is true, so I was quite curious to see what functions have this property. I'm aware that there are probably some very simple solutions, but can anyone think of some more interesting examples?

Answer 1

Note that this means $$u''(x) = u(x) \text{ and }v''(x) = v(x)$$ Hence, we have $$u(x) = ae^x + be^{-x} \text{ and }v(x) = ae^x - be^{-x}$$