= Solution
The steady profile has no time <derivative>. Substituting $u=\widetilde u+u_s$ and using its steady equation cancels the forcing, giving
$$
\boxed{\widetilde u_t=\nu\widetilde u_{xx},\qquad
\widetilde u(0,t)=\widetilde u(1,t)=0}.
$$
The initial condition is
$$
\boxed{\widetilde u(x,0)=-u_s(x)=-\frac{G}{\nu n^2\pi^2}\sin(n\pi x)}.
$$
The 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.
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