can someone help me with the following problem?
Let be $f\in{C^2\mathbb{(R)}}$ , $g\in{C^1\mathbb{(R)}}$. Then $y(t,x)=\frac{f(x+t)+f(x-t)}{2}+\frac{1}{2}\int_{x-t}^{x+t}{g(r)dr}$ is a solution of wave equation with initial conditions ($y(0,x)=f(x)$, $\delta_ty(0,x)=g(x))$ in $\mathbb{R}$. How can we prove the uniqueness of this solution?
I have found some proof of this using energy, but is there a prove which doesn't use it?
Thanks.