1. Consider the general linear second-order ODE Poy" +P1y + P2y = - (1) where po, P1, and p2 are functions of x and A is a constant parameter. The standard form of the Sturm-Liouville ODE (py')' + qy=-Xwy, where p, q, and w are functions of x and A is a constant parameter. (a) Assuming that po, P1, P2, and à are given, multiply Eq. 1 by w. By comparing to Eq. 2, show that the unknown p, q, w, A must satisfy p = wpo, p' = wp1, q = wp2, and λ = A. (b) Show that p = = exp(fdx). (c) Show that w = = -Ay, PO exp(dx). der ODF can in principle he put into the
1. Consider the general linear second-order ODE Poy" +P1y + P2y = - (1) where po, P1, and p2 are functions of x and A is a constant parameter. The standard form of the Sturm-Liouville ODE (py')' + qy=-Xwy, where p, q, and w are functions of x and A is a constant parameter. (a) Assuming that po, P1, P2, and à are given, multiply Eq. 1 by w. By comparing to Eq. 2, show that the unknown p, q, w, A must satisfy p = wpo, p' = wp1, q = wp2, and λ = A. (b) Show that p = = exp(fdx). (c) Show that w = = -Ay, PO exp(dx). der ODF can in principle he put into the
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How do I do question 1.
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1. Consider the general linear second-order ODE
Poy" +P1y' + P2y = −
(1)
where po, P1, and p2 are functions of x and A is a constant parameter. The standard
form of the Sturm-Liouville ODE
(py')'+ qy = -Awy,
where p, q, and w are functions of x and A is a constant parameter.
(a) Assuming that po, P1, P2, and à are given, multiply Eq. 1 by w. By
comparing to Eq. 2, show that the unknown p, q, w, λ must satisfy p =
= wp1, 9= wp2, and λ = A.
wpo, p'
(b) Show that p= exp(fdx).
(c) Show that w = exp(fdx).
1
PO
- Xy,
This shows that a general linear second-order ODE can, in principle, be put into the
standard form of the Sturm-Liouville ODE.
2. For each of the ODEs below, find the weight function w(x) that will allow the
ODE to be written as a standard Sturm-Liouville ODE. Also give the resulting ODE
in the standard form
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Step 1: Here we write the given equations and solve the part - a of the question
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