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?