There should be a list of generic titles for sourceless prob
Posted: Wed Feb 15, 2017 9:13 pm
Find ALL solutions of:
$$ f(x+y) + f(x-y) = 2f(x)cos y $$
$$ f(x+y) + f(x-y) = 2f(x)cos y $$
Let $P(x,y)$ be the FE.
$$P(0,x)\Rightarrow f(x)+f(-x)=2f(0)\cos x=2A\cos x$$
$$P\left (x+\dfrac{\pi}{2}\right )\Rightarrow f(x)+f(x+\pi)=0$$
$$P\left (\dfrac{\pi}{2},\dfrac{\pi}{2}+x\right )\Rightarrow f(-x)+f(x+\pi)=-2f\left (\dfrac{\pi}{2}\right )\sin x=-2B\sin x$$
So, $f(x)=A\cos x+B\sin x$.
Substituting in, the FE works. So we have found the solution.