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(a) Consider the following eigenvalue problem: X" + XX=0 X'(0) = 0, X (π) = 0. (i) Show that all the eigenvalues satisfy >> 0. (ii) Find all eigenvalues and eigenfunctions. + (b) Solve the following wave equation with mixed boundary conditions on an interval. (You can make use of the results obtained in (a).) Utt - c²Uxx = 0 Ur(0, t) = 0, U(π, t) = 0 (U(x, 0) = 0, Ut(x, 0) = 6c. cos (3x).

(a) Consider the following eigenvalue problem: X" + XX=0 X'(0) = 0, X (π) = 0. (i) Show that all the eigenvalues satisfy >> 0. (ii) Find all eigenvalues and eigenfunctions. + (b) Solve the following wave equation with mixed boundary conditions on an interval. (You can make use of the results obtained in (a).) Utt - c²Uxx = 0 Ur(0, t) = 0, U(π, t) = 0 (U(x, 0) = 0, Ut(x, 0) = 6c. cos (3x).

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 12CR
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(a) Consider the following eigenvalue problem: X" + XX = 0 X'(0) = 0, X(T) = 0. (i) Show that all the eigenvalues satisfy > 0. (ii) Find all eigenvalues and eigenfunctions. + (b) Solve the following wave equation with mixed boundary conditions on an interval. (You can make use of the results obtained in (a).) Utt- c²Urx = 0 U₂(0, t) = 0, U(π, t) = 0 U (x, 0) = 0, U₁(x, 0) = 6c cos(x).
Transcribed Image Text:(a) Consider the following eigenvalue problem: X" + XX = 0 X'(0) = 0, X(T) = 0. (i) Show that all the eigenvalues satisfy > 0. (ii) Find all eigenvalues and eigenfunctions. + (b) Solve the following wave equation with mixed boundary conditions on an interval. (You can make use of the results obtained in (a).) Utt- c²Urx = 0 U₂(0, t) = 0, U(π, t) = 0 U (x, 0) = 0, U₁(x, 0) = 6c cos(x).
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