Let 0 < a < b. Consider the functional rb S[y] = [°* dx x³ (y^² – ¾y³). x5 - d²y 5 dy dx² x dx Euler-Lagrange equation for S[y] may be written as + Y1 = + y² = 0. first-integral of S[y] is 4x³yy' + x6 (y^² + ²y³) = c, where c is constant, stating any theorems that you use. Let A and B be constants. By direct substitution or otherwise, show that the functions ?? Ax-2 and y2 = B B (x² + 3/₁1) 24 both give solutions of the first-integral for any A and B, but that only one of y₁ and y2 satisfies the Euler-Lagrange equation for arbitrary A or B.

Let 0 < a < b. Consider the functional rb S[y] = [°* dx x³ (y^² – ¾y³). x5 - d²y 5 dy dx² x dx Euler-Lagrange equation for S[y] may be written as + Y1 = + y² = 0. first-integral of S[y] is 4x³yy' + x6 (y^² + ²y³) = c, where c is constant, stating any theorems that you use. Let A and B be constants. By direct substitution or otherwise, show that the functions ?? Ax-2 and y2 = B B (x² + 3/₁1) 24 both give solutions of the first-integral for any A and B, but that only one of y₁ and y2 satisfies the Euler-Lagrange equation for arbitrary A or B.

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 28E

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