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Homework 8

Math 442: Intro to Partial Differential Equations

(Exercises are taken from Partial Differential Equations An Introduction, Second Edition by Walter A. Strauss.)

1. Exercise §10.2 #1. Show that with the initial conditions (26), all the cos√ λct terms in the series (18) are missing. Also show that Dmm = Cnm = 0 for n = 0.

2. Exercise §10.2 #4. Find all the solutions of the wave equation of the form u = e −iwtf(r) that are finite at the origin, where r = p x 2 + y 2.

3. Exercise §10.5 #7. Find all the solutions u(x) of the ODE xu'' − u' + xu = 0. (Hint: Substitute u= xv.)