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Among all the admissible functions $y = y(x)$, find those that extremise the functional $$J[y] = \int_0^1 (yy')^2dx$$ subject to the constraint $\int_0^1 y^2 dx =3$ and the boundary conditions $y(0)=1, y(1)=2$.

Am I correct in saying denoting $F=(yy')^2 + \lambda y^2$. Thus the extrema corresponding to this problem are the extrema for the functional, $$J[y] =\int_0^1 F dx$$ and therefore they are solutions of the Euler-Lagrange equation $$\lambda y -y^2y''-(yy')^2=0.$$ Could some clarify whether my Euler-Lagrange equation is correct. I am pretty confused as to where to go from here. Any help would be grand.

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    Before you try to analyze the Euler-Lagrange equations, think about the fact that $yy'$ is essentially the derivative of $u = y^2$. So try to rephrase the problem in terms of this $u$ and then think about it. (This is not an exercise in setting up Euler-Lagrange equations, it's an exercise in non-convex calculus of variations).2012-11-25

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