# Calculation of Variations – What to do when the equation of Euler Lagrange is a highly nonlinear ode?

In $$mathbb {R} ^ 3$$, suppose that there is a curve on the X-Y plane $$y (x)$$ defined on $$x in [-a,a]$$ satisfactory:

1. $$y (x) geqslant 0$$;
2. $$y (-a) = y (a) = 0.$$

Turn $$y (x)$$ along the x-axis in $$mathbb {R} ^ 3$$ and get a solid revolution.

Minimize the surface $$A = int _ {- a} ^ {a} 2 pi y sqrt {1+ (y)} 2} mathrm {d} x$$fixed volume $$V = int _ {- a} ^ {a} pi y ^ 2 mathrm {d} x = C$$, for a constant $$C$$.

By standard variational method and multiplier of Lagrange, ($$lambda$$ is constant Lagrange), we get the equation of Euler Lagrange, which is nonlinear.
$$frac {1} {(1+ (y)} ^ 2) ^ {1/2}} + lambda y = frac {yy & # 39; & # 39;} {(1+ (y & # 39;)) ^ 2) ^ {3/2}}$$

My question is how to do with these nonlinear equations. Is it possible to get the minimum $$A$$ without solving $$y$$ explicitly?