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Partial Differential Equations: metal rod

Hi,

Im currently in a Partial Differential Equations Course. Any help would be greatly appreciated! Thanks.

Consider a metal rod of length *l*. At t = 0, half of the rod 0<x<*l*/2 is heated to the temperature T_{0 }and the other half *l*/2 < x<*l*/2 has temperature T =0. Both the left and right ends are insulated. Find Fourier series solution for t>= 0.

a. Write the equation. Identify correct boundary conditions (periodic, Dirichlet, Neumann...). Write down initial condition.

I have u_{t}=ku_{xx} and initial conditions 0<x<l and 0<t<inifinity.

I put it down as a Dirichlet boundary condition but was incorrect.

b. Separate variables and solve the eigenvalue problem.

My work for this is in the attachment.

Not sure what to do after.

c. Find the general solution.

See attachment for my work.

The sin part at the end is incorrect. Not sure why.

d. Find the solution of the initial value problem.