# Does anyone know a solution to this ODE?

1. Jan 21, 2012

### stephanie22

Dear all,

I am wondering if anyone knows the solution to the following nonlinear ODE,
$$\left( -3 + \frac{ f(r) f'(r)}{r} \right)(1+ f'(r)^2 ) + f(r) f''(r) = 0$$
subject to the initial conditions f(R) = f(-R) = f_0.

I have a feeling a closed form solution exists to this ODE because I know a very simple closed form solution to the related ODE
$$\left( 2 + \frac{ f(r) f'(r)}{r} \right)(1+ f'(r)^2 ) + f(r) f''(r) = 0 \,.$$
A solution to this ODE is
$$f(r) = (R^2 - r^2)^{1/2} \,.$$

Thanks in advance for any help!

Steph