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Constaint problem: Minimize T(y)=\int _{0}^{1}\left(y'^{2}+x^{2}\right)\,dx\, s.t. K(y)=\int _{0}^{1}y^{2}\,dx=2\,.

Using the method of Langrange multipliers,

\int _{0}^{1}\left(y'^{2}+x^{2}+\lambda y^{2}\right)\,dx=J(y)\,

If the Euler-Lagrange equations are satisfied then this integral has an extremum.

{\frac  {\partial }{\partial y}}\left[y'^{2}+x^{2}+\lambda y^{2}\right]={\frac  {d}{dx}}{\frac  {\partial }{\partial y'}}\left[y'^{2}+x^{2}+\lambda y^{2}\right]\,

2\lambda y=2y''\,

y''=\lambda y\, so y(x)=c_{1}\cos {\sqrt  {\lambda }}x+c_{2}\sin {\sqrt  {\lambda }}x\,

y(0)=0\, gives c_{1}=0\,

y(x)=c_{2}\sin {\sqrt  {\lambda }}x\,

y(1)=0\, gives 0=c_{2}\sin {\sqrt  {\lambda }}\, so {\sqrt  {\lambda }}=n\pi ,n=1,2,...\,

The solution is

y(x)=c_{2}\sin n\pi x\,

2=\int _{0}^{1}y^{2}\,dx=c_{2}^{2}\int _{0}^{1}\sin ^{2}(n\pi x)\,dx\,

2={\frac  {c_{2}^{2}}{2}}\int _{0}^{1}(1-\cos 2n\pi x)\,dx={\frac  {c_{2}^{2}}{2}}\,

c_{2}=\pm 2\,

Calculus of Variations

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