Consider a finite plate, 10 cm by 30 cm, with two insulated sides, one end at 0∘ and the other at a given temperature T=f(x). Try f(x)=100∘; f(x)=x. You should convince yourself that this problem cannot be done using just the solutions (2.7). To see what is wrong, go back to the differential equations (2.5) and solve them if k=0. You should find solutions x, y, xy, and constant [the constant is already contained in (2.7) for k=0, but the other three solutions are not]. Now go back over each of the problems we have done so far and see why we could ignore these k=0 solutions; then including the k=0 solutions, finish the problem of the finite plate with insulated sides.
For the case f(x)=x, the answer is:
T=61(30−y)−π240oddn∑n2sinh3nπ1sinh10nπ(30−y)cos10nπx.