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Question 22.17: Determine the minimum value that the integral J = ∫^1 0 [x^4...

Determine the minimum value that the integral

J=\int _{ 0 }^{ 1 }{ \left[ { x }^{ 4 }{ ({ y }^{ \prime \prime }) }^{ 2 }+4{ x }^{ 2 }{ ({ y }^{ \prime }) }^{ 2 } \right] } dx

can have, given that y is not singular at x = 0 and that y(1) = y′(1) = 1. Assume that the Euler–Lagrange equation gives the lower limit and verify retrospectively that your solution satisfies the end-point condition

{ \left[ \eta \frac { ∂ F }{ ∂ { y }^{ \prime } } \right] }_{ a }^{ b }=0,

where F = F({y}^{\prime}, y, x) and η(x) is the variation from the minimising curve.

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