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Find the extremals of \int _{a}^{b}{\frac  {y'^{2}}{x^{3}}}\,dx\,.

The Euler-Lagrange equations are in general for the integrand functional L\,

{\frac  {\partial L}{\partial y}}-{\frac  {d}{dx}}{\frac  {\partial L}{\partial y'}}+{\frac  {d^{2}}{dx^{2}}}{\frac  {\partial L}{\partial y''}}-{\frac  {d^{3}}{dx^{3}}}{\frac  {\partial L}{\partial y'''}}+...+(-1)^{k}{\frac  {d^{k}}{dx^{k}}}{\frac  {\partial L}{\partial y^{{(k)}}}}=0\,

There is no y\, in the integrand so in this problem the EL-equation is

-{\frac  {d}{dx}}{\frac  {\partial L}{\partial y'}}=0\,

So

{\frac  {d}{dx}}{\frac  {2y'}{x^{3}}}=0\,

Integrate with respect to x\,.

{\frac  {2y'}{x^{3}}}=c_{1}\, a constant.

Prepare to integrate with respect to x\,.

y'={\frac  {c_{1}x^{3}}{2}}\,

Integrate. The solution is

y(x)={\frac  {c_{1}x^{4}}{8}}+c_{2}\,

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