Graduate Difficult partial differential Problem

[big dream]

Problem:
$${\frac {\partial }{\partial t}}A\left( y,t \right) +6\,\Lambda\,\Omega\, \left( {y}^{2}-y \right) \sin \left( t \right) ={\frac {\partial ^{2}}{\partial {y}^{2}}}A \left( y,t \right)$$
$${\frac{\partial }{\partial y}}A \left( t,0 \right) ={\frac {\partial }{\partial y}}A \left( t,1 \right) =0$$
Boundary condition
$${\frac{\partial }{\partial y}}A \left( t,0 \right) ={\frac {\partial }{\partial y}}A \left( t,1 \right) =0$$
ANSWER OF THIS EQUATION IS
$$A \left( y,t \right) =6\,\Lambda\, \left( \Im \right) \, \left\{ [{\frac {i\sinh \left( \alpha\,y \right) }{\alpha}}-{\frac {i \left( 1-\cosh \left( \alpha \right) \right) \cosh \left( \alpha\,y \right) }{\alpha\,\sinh \left( \alpha \right) }}+i{y}^{2}-iy+2\,{\Omega}^{-1}]{{\rm e}^{it}} \right\}$$
Where, $$\alpha=1/2\, \left( 1+i \right) \sqrt {2}\sqrt {\Omega}$$
attempt at a solution

Maple didn't give an answer. I don't know how to get this kind of solution.

 

[bigfooted]

This is a diffusion equation with some periodic source term. You should try separation of variables for this. There is an example on the wiki page, but searching for separation of variables of nonhomogeneous heat equation will lead to many extensive examples.
You could start here:
https://en.wikipedia.org/wiki/Separation_of_variables#Example:_nonhomogeneous_case
http://www.math.psu.edu/wysocki/M412/Notes412_10.pdf

 

[big dream]

Thank you, sir. I got it. This thread can be closed now.