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A necessary condition for the functional defined by X2 = dx f(x, Y(x),Y'(x)) to have a local minimum for ôf of dx Y(x) = y(x) is h(x) + h'(x) | = 0 for all ду lifferentiable functions h with h(x) h(x,) = 0. ) Show how to obtain the Euler-Lagrange condition ôf d of 0 from this. ду dx ôy'

A necessary condition for the functional defined by X2 = dx f(x, Y(x),Y'(x)) to have a local minimum for ôf of dx Y(x) = y(x) is h(x) + h'(x) | = 0 for all ду lifferentiable functions h with h(x) h(x,) = 0. ) Show how to obtain the Euler-Lagrange condition ôf d of 0 from this. ду dx ôy'

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A necessary condition for the functional J defined by J(Y) = | dx f(x,Y(x),Y'(x)) to have a local minimum for af h(x) + ôf h'(x) = 0 for all Y(x) = y(x) is differentiable functions h with h(x,) = h(x,) = 0. (a) Show how to obtain the Euler-Lagrange condition se = 0 from this. d - ду dx ôy' (b) If ƒ does not depend explicitly on x show that af f - y is a constant.
Transcribed Image Text:A necessary condition for the functional J defined by J(Y) = | dx f(x,Y(x),Y'(x)) to have a local minimum for af h(x) + ôf h'(x) = 0 for all Y(x) = y(x) is differentiable functions h with h(x,) = h(x,) = 0. (a) Show how to obtain the Euler-Lagrange condition se = 0 from this. d - ду dx ôy' (b) If ƒ does not depend explicitly on x show that af f - y is a constant.
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