Consider the 1-dimensional heat equation u_t = u_{xx} over the x-interval [0,Pi] with zero end temperatures and initial temperature u(x,0)=sin(3x)-sin(5x).   
The solution u(x,t) given by separation of variables is

`u(x,t)` = exp(-9*t)*sin(3*x)-exp(-25*t)*sin(5*x)

Below is the plot of this solution as a function of x in [0,Pi] for time t in [0,0.8]. Note that the steady-state temperature is zero.

[Plot]