$f(x)=\sin x$ is a well known function which satisfies the differential equation
$$\frac{df}{dx}=f(\pi/2-x)$$ with the initial condition $f(0)=0$; I am just curious to know are these all? If not,what kind of properties functions should satisfy?
My attempt:
Putting $y=\pi/2-x$ does not seem useful to me.
On
Since $\dfrac{df}{dx}=f(\pi/2-x)$, we have $\dfrac{d^2f}{dx^2}=-f(x)$. $$2\dfrac{d^2f}{dx^2}\dfrac{df}{dx}+2\dfrac{df}{dx}f=0$$ We deduce that $$(\dfrac{df}{dx})^2+f^2=C^2$$ Suppose that $C$ is a real number, then we have $$\dfrac{df}{\sqrt{C^2-f^2}}=dx$$ After integrating both sides, we deduced that $$f(x)=C\sin (x+\phi)$$
If $f'(x)=f(\frac {\pi} 2 -x)$ then $f''(x)=-f(x)$ since $\frac {\pi} 2 -(\frac {\pi} 2 -x)=x$. The only solutions of $y''=-y$ are of the form $a\sin x+b \cos x$. From the condition $f(0)=0$ we must have $f(x)=a\sin x$ for some constant $a$.