What are the eigenvalues and eigenfunctions: x′′ + λx = 0, x(1) = x(3), x′(1) = x′(3) - Definey by x(z) = y((z−2)π) for 1 ≤ z ≤ 3. Show that y(−π) = y(π) and y′(−π) = y′(π) - Substitute into the equation to get y′′((z−2)π) + (λ/π2) y((z−2)π), for 1 ≤ z ≤ 3 - Use the change of variable t = (z − 2)π to show that the above equation has a non-zero solution if and only if either λ = k2π2 for some integer k ≥ 1 or λ = 0 and the solutions (eigenfunctions) are given by cos(kt), sin(kt) and 1 for −π ≤ t ≤ π. - Plug back t = (z − 2)π to find the eigenfunctions and eigenvalues of original equation

What are the eigenvalues and eigenfunctions: x′′ + λx = 0, x(1) = x(3), x′(1) = x′(3) - Definey by x(z) = y((z−2)π) for 1 ≤ z ≤ 3. Show that y(−π) = y(π) and y′(−π) = y′(π) - Substitute into the equation to get y′′((z−2)π) + (λ/π2) y((z−2)π), for 1 ≤ z ≤ 3 - Use the change of variable t = (z − 2)π to show that the above equation has a non-zero solution if and only if either λ = k2π2 for some integer k ≥ 1 or λ = 0 and the solutions (eigenfunctions) are given by cos(kt), sin(kt) and 1 for −π ≤ t ≤ π. - Plug back t = (z − 2)π to find the eigenfunctions and eigenvalues of original equation

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.1: Eigenvalues And Eigenvectors

Problem 65E

See similar textbooks