Intergration, Functional and variations.Block III, Consider the functional S[y]=?

Intergration, Mathematics

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Functional and variations.Block III,
Consider the functional
S[y]=?_1^2 v(x^2+y'')dx , y(1)=0,y(2)=B
Show that if ?=S[y+eg]-S[y], then to second order in e,

?=1/2 e?_1^2¦?g^''/v(x^2+y^'') dx-1/8 e^2 ?_1^2¦??g^''?^2/(x^2+y'')^(3/2) dx??

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Intergration, Mathematics

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