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We begin by finding the EL equation for the auxiliary functional S[y] = = - S₁ 12 dx (xy2+ Ay - λy) = = · L dx F ( dxF(x,y,y', 2). d 0 = (Fy') - Fy (2xy') – (A – 1). dx dx Integrating gives A -λ B y' + 2 x and integrating again gives A - λ y = -x+Blogx+C, 2 77 come where C is another constant. The initial condition y(1) = 0 gives C = -(A - λ)/2. We now apply the natural boundary condition (HB pg 24) given by Fy = 0 at x = v. That is 0 = Fy | x=v = 2xy' | x=v' that is y'(v) = 0. Now A - X 0 = y' (v) = إن B v(A - λ) + hence B = 2 v Hence A - λ y(x) = K(x − 1 - vlogx) where K = 2

We begin by finding the EL equation for the auxiliary functional S[y] = = - S₁ 12 dx (xy2+ Ay - λy) = = · L dx F ( dxF(x,y,y', 2). d 0 = (Fy') - Fy (2xy') – (A – 1). dx dx Integrating gives A -λ B y' + 2 x and integrating again gives A - λ y = -x+Blogx+C, 2 77 come where C is another constant. The initial condition y(1) = 0 gives C = -(A - λ)/2. We now apply the natural boundary condition (HB pg 24) given by Fy = 0 at x = v. That is 0 = Fy | x=v = 2xy' | x=v' that is y'(v) = 0. Now A - X 0 = y' (v) = إن B v(A - λ) + hence B = 2 v Hence A - λ y(x) = K(x − 1 - vlogx) where K = 2

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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We begin by finding the EL equation for the auxiliary functional S[y] = = - S₁ 12 dx (xy2+ Ay - λy) = = · L dx F ( dxF(x,y,y', 2). d 0 = (Fy') - Fy (2xy') – (A – 1). dx dx Integrating gives A -λ B y' + 2 x and integrating again gives A - λ y = -x+Blogx+C, 2 77 come where C is another constant. The initial condition y(1) = 0 gives C = -(A - λ)/2. We now apply the natural boundary condition (HB pg 24) given by Fy = 0 at x = v. That is 0 = Fy | x=v = 2xy' | x=v' that is y'(v) = 0. Now A - X 0 = y' (v) = إن B v(A - λ) + hence B = 2 v Hence A - λ y(x) = K(x − 1 - vlogx) where K = 2
Transcribed Image Text:We begin by finding the EL equation for the auxiliary functional S[y] = = - S₁ 12 dx (xy2+ Ay - λy) = = · L dx F ( dxF(x,y,y', 2). d 0 = (Fy') - Fy (2xy') – (A – 1). dx dx Integrating gives A -λ B y' + 2 x and integrating again gives A - λ y = -x+Blogx+C, 2 77 come where C is another constant. The initial condition y(1) = 0 gives C = -(A - λ)/2. We now apply the natural boundary condition (HB pg 24) given by Fy = 0 at x = v. That is 0 = Fy | x=v = 2xy' | x=v' that is y'(v) = 0. Now A - X 0 = y' (v) = إن B v(A - λ) + hence B = 2 v Hence A - λ y(x) = K(x − 1 - vlogx) where K = 2
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