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Supplementary Problems
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Exercise 111. Solve the boundary problem u” = 0 for 0 < x < 1 with u (0) +
0
ku(0) = 0 and u (1) ± ku(1) = 0. Do the + and − cases separately. What is
special about the case k = 2?
Exercise 112. Consider the Neumann problem ∆u = f(x, y, z) in D, ∂u = 0 on
∂n
boundary D. What can we surely add to any solution to get another solution? So
we don’t have uniqueness.
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