Consider the functional S[y] = [ dz 3x^¼y³y³. S[y] is invariant under the scale transformation ĩ = ax, y = ßy if aß³ = 1. first-integral of S[y] is x¹y¹y'² + 2x³y³y'³ = c = constant, solution of this equation is A6 y = Ax-1/3 where c = 27 Euler-Lagrange equation for S[y] may be written 12 xy”+xy+2y = 0. By differentiating the first-integral, show that if y satisfies the first-integral then 2 (xyy" + xy'² +2yy')(y + 3xy') = 0.

Consider the functional S[y] = [ dz 3x^¼y³y³. S[y] is invariant under the scale transformation ĩ = ax, y = ßy if aß³ = 1. first-integral of S[y] is x¹y¹y'² + 2x³y³y'³ = c = constant, solution of this equation is A6 y = Ax-1/3 where c = 27 Euler-Lagrange equation for S[y] may be written 12 xy”+xy+2y = 0. By differentiating the first-integral, show that if y satisfies the first-integral then 2 (xyy" + xy'² +2yy')(y + 3xy') = 0.

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.

Chapter9: Multivariable Calculus

Section9.2: Partial Derivatives

Problem 33E

See similar textbooks