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Consider this problem: r²y" + xy/ + ? = - 0, y(1) = 0 y(2) = 0. a) Place the differential equation in standard Sturm-Liouville form, showing the steps used to do so.

Consider this problem: r²y" + xy/ + ? = - 0, y(1) = 0 y(2) = 0. a) Place the differential equation in standard Sturm-Liouville form, showing the steps used to do so.

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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Consider this problem: 2²y" + ry/ + = 0, y(1) = 0 y(2) = 0. a) Place the differential equation in standard Sturm-Liouville form, showing the steps used to do so. b) Identify and state the weighting function. c) Are the eigenvalues of this problem real? d) State the orthogonality condition for the eigenfunctions of the problem. You do not need to solve for the eigenfunctions, nor do you need to state the eigenvalues.
Transcribed Image Text:Consider this problem: 2²y" + ry/ + = 0, y(1) = 0 y(2) = 0. a) Place the differential equation in standard Sturm-Liouville form, showing the steps used to do so. b) Identify and state the weighting function. c) Are the eigenvalues of this problem real? d) State the orthogonality condition for the eigenfunctions of the problem. You do not need to solve for the eigenfunctions, nor do you need to state the eigenvalues.
Expert Solution
Step 1

The differential equation given here is

x2y''+xy'+λx=0

y''+y'x+λx3=0

compare it with equation

y''+y'px+qxy=0

P'xPx=px ⇒Px =ce∫pxdx

Px =c e∫1xdx =cx

let c= 1 and λw-Q =Pxqx

 λw-Q=cxλ/x3

So Q = 0 and w =1/x2

So, Sturm - Liouville form is given as

-∂∂xpx∂y∂x+qyy=wλy

plugging the values,

-∂∂xcx∂y∂x+λx3y=1x2λy

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