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.