Problem 2 (eigenfunction expansion, Haberman 3.4.11) Consider v(x, t) = (1-e */4) sin(#x/2) and w(r,t) = -esin(mr) on (0, 2) x (0, co). (a) Find the initial/boundary value problem (for a forced heat equation) satisfied by (b) Find the initial/ boundary value problem satisfied by w. (c) Find the initial/ boundary value problem satisfied by v + w. (d) Find a point r E (0, 2) at which the temperature v( r, t) tw(r, t) first decreases and then increases. Find a point r e (0, 2) at which the temperature v(r, t) + w(x, t) only increases. Can you increase the forcing to ensure the temperature at all points only increases? Hint(s): This means to consider av(r, t) + w(x, t) for a > 1. If you're stuck you might want to look at part (e) first and then come back to this part