I want to use the Euler equation to find $y=y(x)$ to get the extremals for the following functional: $$\int_a^b \left[xy + 2\left(\frac{dy}{dx}\right)^2\right] \, \mathrm{d}x$$

I know that the Euler equation is given as $$\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'} = 0$$ but I am not sure how to apply it in this situation.

Is it even possible to find extremals without boundary conditions?


Calling $L(y, y', x) = x y +2y'^2$ The Euler equation is

$$ L_y -(L_{y'})' = 4y''-x=0 $$

Solving for $y$ we have

$$ y = C_1+x C_2 +\frac{x^3}{24} $$

The boundary conditions are fixed using the constants $C_1, C_2$


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