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 For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter4: Calculating The Derivative

Section4.5: Derivatives Of Logarithmic Functions

Problem 47E

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