i have to use method of separation of variable to solve the wave equation below. I have no idea where to start the calculation. please help~!!
If you are given, $\displaystyle \frac{\partial^2 u}{\partial t^2} = k^2 \frac{\partial^2 u}{\partial x^2}$ ($\displaystyle k>0$)
With boundary conditions $\displaystyle u(0,t) = u(\pi,t) = 0$.
And with initial conditions $\displaystyle u(x,0) = f(x) $ and $\displaystyle u_t(x,0) = g(x)$.
Where $\displaystyle f(x) = \sum_{n=1}^N b_n \sin (nx)$ and $\displaystyle g(x) = \sum_{n=1}^N a_n \sin (nx)$.
The solution is given by,
$\displaystyle u(x,t) = \sum_{n=1}^N \left[ \frac{a_n}{kn} \sin (nt) + b_n \cos (nt) \right] \sin (nx)$