Consider the wave equation $$\frac{\partial ^2 u}{\partial t^2}= c^2 \frac{\partial ^2 u}{\partial x^2}$$ where $u$ is the vertical height of the string. Given the following boundary conditions $u(0,t) = u(L,t) = 0$ and initial conditions $u(x,0) = f(x)$ , $\frac{\partial u}{\partial t} (x, 0) = g(x)$.
Prove that the above system has a unique solution.
I was told to use conservation of energy to solve this problem, that is $d E/ d t = c^2( u_x(L,t) u_t (L,t) - u_x(0,t) u_t (0,t))$
I went with the standard approach of assuming two solutions $v, w$ to the above system, and tried to arrive to a contradiction. I am trying to achieve a stage where I can get to $v-w=0$, but I can't seem to get there. Simply plugging and subtracting in the PDE and the initial conditions doesn't seem to work. I am unsure of how to use energy conservation here. I know how to get to the solution using separation of variables, and considering positive, negative and zero eigenvalues and using Fourier series to get to a solution. Though, I doubt the question is asking for that... Any advice would be appreciated.