Problem 2 ( eigenfunction expansion, Haberman 3.4.11) Consider v(x, t) = (1 -e-*2t/4 ) sin(x/2) and w(x, t) = -esin(NI) on (0, 2) x (0, co). (a) Find the initial/boundary value problem (for a forced heat equation) satisfied by U. (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)+w(x, t) first decreases and then increases. Find a point x E (0, 2) at which the temperature v(x, 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. (e) Animate v + w. Explain how v + w relates to your discussion in part (e) of Problem 1