Answered: 4. For the Sturm-Liouville problem y" +… | bartleby

Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 1BEXP

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4. For the Sturm-Liouville problem
y" + xy = 0, y(0) = 0, y'(L) = 0
find the eigenvalues and eigenfunctions. Verify orthogonality.

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5.Consider the eigenvalues given by equation (39). Show that (σ1X+σ2Y)2−4(σ1σ2−α1α2)XY=(σ1X−σ2Y)2+4α1α2XY.σ1X+σ2Y2−4σ1σ2−α1α2XY=σ1X−σ2Y2+4α1α2XY. Hence conclude that the eigenvalues can never be complex-valued.
Prove that the eigenvalue problem y''+ (lamba)y=0, y(0)=0, y(1)+y'(1)=0 has no negative eigenvalues. (Suggestion: Show graphically that the only root of the equation tanhz=-z is z=0.)
I try to use the eigenvalue to find eigenvector, and I've already found the relation between x and y. I can substitute x=1 and get y=-i, but the solution provided is substituting x=-1 and get y=i. There is a problem, how can I know which value of x should I choose?
Consider the eigenvalue problem y"+2y'+(lambda)y=0; y(0)=0, y'(1)=0. Show that the eigenvalues are all positive and that the nth positive eigenvalue is (lambda)n= an^2+1 with associated eigenfunction yn(x)= e^-x sin(anx) where anx is the nth positive root of tanz=z
Find the Eigenvalues and Eigenfunctions of the following Sturm_Liouville problems posed.y´´ +2y´ + λy=0 with boundary conditions y´(0)= y´(1)=0
Find the Eigenvalues and Eigenfunctions of the following Sturm_Liouville problems posed.y´´ + λy=0 with boundary conditions y´(0)=y´(1)=0
In the Sturm-Liouville problem (xy')'+ (lambda/x)y =0 ,y(1) =0 y(e)=0 what is the eigenvalues and eigenfunction.

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