Discontinuous Solution for Calculus of Variations?

[arhez]

For the integral J =\int f(y,y_x,x) dx

if
f(y,y_x,x) = y^2(x)

find a discontinuous solution similar to the Goldschmidt solution.

This is the first time I approach the calculus of variations, so I thought of using the Eulere
equation f - y_x\frac{\partial f}{\partial y_x}

But I end up with y^2=C. So I am missing something, any hint as to what that might be, is appreciated. Thanks.

 

[lippyka]

The Goldschmidt solution for this particular case is given by: J= \int y^2 dx + C,where C is a constant.