Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 20 / b

The steady profile has no time derivative. Substituting $u=\tilde{u}+u_s$ and using its steady equation cancels the forcing, givingThe initial condition isThe resulting heat equation is independent of $G$, as are its boundary conditions. Its initial amplitude still depends on $G$; removing a source from the evolution equation does not remove it from the whole problem.