Heat Flow Problem

1. The problem statement, all variables and given/known data

Find a formal solution to the given initial boundary value problem.

du/dt=5(d^2u/dx^2) 0<x<1 t>0
u(0,t)=u(1,t)=0 t>0
u(x,0)=(1-x)(x^2) 0<x<1

2. Relevant equations

1) u(x,t) = a₀/2 + sum[a_n*e^(-b(n pi/L)^2*t) * cos(n pi x/L)

2) Fourier series equation

3. The attempt at a solution

(1-x)(x^2) = a₀/2 + sum(a_n * cos(n pi x) with c_n = a_n

I calculate a₀=1/6

a_n = 2* integral[(1-x)(x^2)(cos n pi x)dx] from 0 to 1


I'm wondering if this is write so far? And if so, how do I proceed from here? Do I just plug everything back into the general u(x,t) equation?

Thanks!