a) Consider the following eigenvalue problem: X′′ + λX = 0 X′(0) = 0,X(π) = 0. (i) Show that all the eigenvalues satisfy λ > 0. (ii) Find all eigenvalues and eigenfunctions.
a) Consider the following eigenvalue problem: X′′ + λX = 0 X′(0) = 0,X(π) = 0. (i) Show that all the eigenvalues satisfy λ > 0. (ii) Find all eigenvalues and eigenfunctions.
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′′ + λX = 0
X′(0) = 0,X(π) = 0.
(i) Show that all the eigenvalues satisfy λ > 0.
(ii) Find all eigenvalues and eigenfunctions.
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b) Solve the following wave equation with mixed boundary conditions on an interval. (You can make use of the results obtained in (a).)
Utt − c^2Uxx = 0
Ux(0,t) = 0,
U(π,t) = 0
U (x, 0) = 0,
Ut (x, 0) = 6c · cos( 23 x)
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Transcribed Image Text:(a) Consider the following eigenvalue problem:
X" + XX
X'(0) = 0, X (π) = 0.
=
(i) Show that all the eigenvalues satisfy > > 0.
(ii) Find all eigenvalues and eigenfunctions.
0
(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
U₂(0, t) = 0, U(Tπ, t) = 0
U(x, 0) = 0, Ut(x, 0) = 6c. cos(x).
0
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