How would you attack this equation?(adsbygoogle = window.adsbygoogle || []).push({});

(1/2)y'' + y^2 = 1

I'm trying to find an approximate solution for the trajectory of a ball acted on by constant gravity acceleration and quadratic drag (proportional to the speed squared), and it turns out the speed squared satisfies an equation that is equivalent to this one after non-dimensionalizing. Any kind of solution would be nice, whether it is good for large x, small x, or whatever.

I was thinking that for x << 1, the equation would be approximately (1/2)y'' + y^2 = 0, in which case y = -3/x^2 is a solution. But then I don't know how to proceed from there to a better approximation. For x >> 1, I was thinking y^2 = 1 would be a first approximation, but again I don't know how to get any farther than that.

thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# (1/2)y'' + y^2 = 1

**Physics Forums | Science Articles, Homework Help, Discussion**