Question

2.29 Determine the eigenvalues and eigenfunctions of the problem [(x + 3)^2 y']' + ? y = 0, -2 ? x ? 1, y(-2) = y(1) = 0. 2.30 Let u'' + ? u = 0, 0 ? x ? ?, u(0) = 0, 2u(?) - u'(?) = 0. (a) Show how the positive eigenvalues ? may be determined. Are there any eigenvalues in (-?, 0]? (b) What are the corresponding eigenfunctions?

          2.29 Determine the eigenvalues and eigenfunctions of the problem
[(x + 3)^2 y']' + ? y = 0, -2 ? x ? 1,
y(-2) = y(1) = 0.
2.30 Let
u'' + ? u = 0, 0 ? x ? ?,
u(0) = 0, 2u(?) - u'(?) = 0.
(a) Show how the positive eigenvalues ? may be determined. Are there any eigenvalues in (-?, 0]?
(b) What are the corresponding eigenfunctions?

2.29 Determine the eigenvalues and eigenfunctions of the problem
[(x + 3)^2 y']' + ? y = 0, -2 ? x ? 1,
y(-2) = y(1) = 0.
2.30 Let
u” + ? u = 0, 0 ? x ? ?,
u(0) = 0, 2u(?) - u'(?) = 0.
(a) Show how the positive eigenvalues ? may be determined. Are there any eigenvalues in (-?, 0]?
(b) What are the corresponding eigenfunctions?

Added by Robert H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

04/01/2023

Step 1

We can rewrite the differential equation as: \[y'' + \frac{A}{(z + 3)^2} y = 0\] Let's assume a solution of the form: \[y(z) = e^{mz}\] Plugging this into the differential equation, we get: \[m^2 e^{mz} + \frac{A}{(z + 3)^2} e^{mz} = 0\] Dividing by Show more…

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Determine the eigenvalues and eigenfunctions of the problem [(x + 3)^2y']' + ̇λ y = 0, -2 ≤ x ≤ 1, y(-2) = y(1) = 0. Let u'' + λ u = 0, 0 ≤ x ≤ π, u(0) = 0, 2u(π) - u'(π) = 0. (a) Show how the positive eigenvalues λ may be determined. Are there any eigenvalues in (-∞, 0]? (b) What are the corresponding eigenfunctions?