If $\frac{df(x)}{dx} = e^{-x}\,f(x) + e^{x}\,f(-x)$, then is $f(x)$ even or odd?

Question

If $\frac{df(x)}{dx} = e^{-x}\,f(x) + e^{x}\,f(-x),$ then is $f(x)$ is even or odd? ($f(0) =0$ is given)

Clearly $\frac{df(x)}{dx}$ is an even function. How can I check the nature of $f(x)$ from this?

Answer 1

Note that $$f(x)=\int_0^xf'(u)du$$therefore $$f(-x){=\int_0^{-x}f'(u)du\\=\int_0^x f'(-u)d(-u)\\=-\int_0^xf'(u)du\\=-f(x)}$$hence $f(x)$ is odd.