# Method of separation of variables for wave equation

1. Apr 25, 2013

### sigh1342

1. The problem statement, all variables and given/known data
$$u_{tt} = a^2u_{xx} , 0<x< l , t>0 ,$$a is constant
$$u(x,0)=sinx , u_{t} (x,0) = cosx , 0<x< l , t>0$$
$$u(0,t)=2t , u(l,t)=t^2 , t>0$$

2. Relevant equations

3. The attempt at a solution
I can solve the eigenvalue problem of X(x), and then solve for T(t), but I dont know how to solve the initial value problem for $$u(x,0) =sinx , and u_{t} (x,0) = cosx$$
with I can only compute the fourier expansion of $$sinx$$ and $$cosx$$ with $$λ_{n} = \frac { (n \pi)^2} {l^2}$$ , but the ans looks like ugly, and compare term is fail.
by the way I'm sorry for my poor english.