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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=-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

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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5°F unny 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 Q Search
Transcribed Image Text:5°F unny 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 Q Search
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