Question: Problem 2 (eigenfunction expansion, Haberman 3.4.11) Consider (x,t) = (1-e-) sin(xx/2) and on (0.2) x (0, co). (n) Find the initial/boundary value problem (for a

Problem 2 (eigenfunction expansion, Haberman 3.4.11) Consider (x,t) = (1-e-") sin(xx/2) and on (0.2) x (0, co). (n) 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 e + w. (d) Find a point a C (0, 2) at which the temperature v(x, () +w(x, () first decreases and then increases. Find a point a E (0, 2) at which the temperature r(r, f) + w(r, !) only increases. Can you increase the forcing to ensure the temperature at all points only increases? Hint(s): This means to consider ar(s, f) + w(r. () for a > 1. If you're stuck you might want to look at part (e) first and then come back to this part

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