How many types of functions $f(x)$ satisfy $(f(x))^2$ + $(f'(x))^2 = constant$?

Question

I know of two types of $f(x)$ that works. First one is if $f(x) = c$, of course. Second one is if it's a trigonometric function: $f(x) = c * \cos(x)$ or $f(x) = c * \sin(x)$.

My question is, are there any other types of functions that satisfy this criteria?

Answer 1

Differentiating both sides:

$$2ff'+2f'f''=0$$ $$2f'(f+f'')=0$$

So either $f'=0$, which gives $f(x)=c$, or $f+f''=0$, which gives $f(x)=A\sin(x)+B\cos(x)$.

Answer 2

Hint.

$$ f'(x) = \pm\sqrt{c-f(x)^2} $$

then

$$ dx =\pm\frac{df}{\sqrt{c-f^2}} $$

giving

$$ x + C_0 = \pm\arctan\left(\frac{f}{\sqrt{ c-f^2}}\right) $$

etc.