Separation of variables help please!

Question

The temperature distribution in a semi-infinite rod follows the diffusion equation

k(∂²u/∂x²)= ∂u/∂t

The temperature of the rod at x=0 is varied (relative to temperature T₀) as u(0,t) = ΔTsin(wt)

By using separation of variables with an imaginary separation constant show that the solution is, for x>=0

u(x,t) = ΔT exp(-x*sqrt(w/2k))sin(wt + x*sqrt(w/2k))

Could someone help me in detail as I can never seem to get the final answer...

Answer 1

 I thought my solution may still be helpful although I am missing the negative, so I wrote down the major steps as a guideline for you.  Please go to the following link to access the file.  

 

 http://www.wyzant.com/resources/files/346387/separation_of_variables