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# Consider the differential equation d2 dx2 + = 0. Determine the eigenvalues (and

ISBN: 9780321797056 284

## Solution for problem 2.3.3 Chapter 2.3

Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition

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Problem 2.3.3

Consider the differential equation d2 dx2 + = 0. Determine the eigenvalues (and corresponding eigenfunctions) if satisfies the following boundary conditions. Analyze three cases ( > 0, = 0, < 0). You may assume that the eigenvalues are real. (a) (0) = 0 and ()=0 *(b) (0) = 0 and (1) = 0 (c) d dx(0) = 0 and d dx(L) = 0 (If necessary, see Section 2.4.1.) *(d) (0) = 0 and d dx(L)=0 (e) d dx(0) = 0 and (L)=0 *(f) (a) = 0 and (b) = 0 (You may assume that > 0.) (g) (0) = 0 and d dx(L) + (L) = 0 (If necessary, see Section 5.8.)(a) u(x, 0) = 6 sin 9x L (b) u(x, 0) = 3 sin x L sin 3x L (c) u(x, 0) = 2 cos 3x L (d) u(x, 0) = 1 0 < x L/2 2 L/2

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