Let $\lambda$ be a nonzero real constant. Find all functions $f,g: \Bbb R \rightarrow \Bbb R$ that satisfy the functional equation $f(x+y)+g(x−y)=\lambda f(x)g(y)$.
I try this :
Let $y=0$ in the equation to get $f(x)+g(x)=\lambda f(x)g(0)$
Here We have two cases:
$g(0)=0$ here $f(x)=-g(x)$
$g(0) \neq 0$ here $g(x)=\beta f(x)$ , $\beta = g(0)\lambda-1$
Is this true? And if this true how I can complete the the solution especially in case two?