Partial diff equation

It is given that that a Fourier Cosine series of teh function f(x) is given by (see file)
A PDE is given by:

u_t = u_xx 0<x>Pi t>0

With boundary conditions:

u_x(0,t) = u_x(pi, t)=0

1. Determine all the solutions of the PDE that satisfy these boundary conditions.
2. find teh solution that satisfies in addition the initial condition u(x,0) = f(x)

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