Differential Equations Math 285 Hour Exam 3 Fall, 2011

1.	Let f(t) = t - 1 where 0 < t < 3 . a. Find the Fourier sine series of f(t). b. Give a careful graph of the sum of the series on the interval (-6, 6). c. Differentiate the series term wise. Is the resulting equality correct for 1 < t < 2? ∞ Answers: a. f(t) ~ (-2/π) Σ bn sin nπt /3 where bn = (-2/n) [ (-1)^n + 1 ] n = 1 b. c. No.
2.	Consider the following differential equation. m’’ + x = Σ (1 / π n^2) sin 2nt where x is a function of t on the entire real line. n odd a. For which m from the following list does pure resonance occur? (Several may fit.) m = ½, 1/8, 1 / 18, 1 / π, 1 / 9π b. Find a solution of the equation for m = 1. Answers: a. None b. x = Σ [ 1/ (πn^2 (1 – 4n^2) sin 2nt n odd Click here to continue with this exam.
3.	Consider the eigenvalue problem: y” + λ y = 0 y ‘(0) = 0, y(5π) = 0 Assume as given that there are no negative eigenvalues. a. Is λ = 0 is an eigenvalue? b. Find all positive eigenvalues and the corresponding eigenfunctions. Answers: a. No b. λn = n^2 / 100, yn = cos (nx/10) Click here to continue with this exam.

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