The one you have the the simplest one (1 dimensional):
Given this PDE with the Boundary Value Problem (fixed-fixed):
The solution is given by:
That is the coefficients of the half-range Fourier sine expansion.
What I wrote above is the solution to this problem (which was derived by Seperation of Variables.)
In you equation we have,
(that is the length of the rod).
(the coefficient in front).
So the solution is given by,
Can you take if from there?