Find all functions $f \in C^{1}(\mathbb{R})$ such that$$f(x)^{2} = \int_{0}^{x} (f(t)^{2}+f'(t)^{2}) dt + 2022$$for all $x$ real.
My attempt: If $f \in C^{1}(\mathbb{R})$ is a solution, then$$2f(x)f'(x) = f(x)^{2}+f'(x)^{2}.$$Thus$$0 = f(x)^{2}-2f'(x)f(x)+f'(x)^{2} = (f(x)-f'(x))^{2}.$$This implies that $f(x) = f'(x)$ for all $x \in \mathbb{R}$, so $f(x) = ce^{x}$ for some $c \in\mathbb{R}$.I don´t know why but when I substitute the solution I don´t get the equality. Is my reasoning correct? Should I justify why $f$ has to be a scalar multiple of the exponential map?