CoV8

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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\,

Main Page : Calculus of Variations

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