Okay, I have this not so pretty 2nd order non-linear ODE I should be able to solve numerically.
$$f''(R) + \frac{2}{R} f'(R)=\frac{0.7}{R} \left( \frac{1}{\sqrt{f(R)}} - \frac{0.3}{\sqrt{1-f(R)}} \right),$$
$$f(1)=1.$$
The function around the origin is behaving very wildly.
I was thinking of breaking this guy up into a system of two first order ODE's and then solve, but I have no idea how to set this up. What method should I use to set up the system of ODE's?
If there is some other method rather than numerically solving a system of differential equations, please feel welcome to share. Thanks.
The general method to reinterpret a higher-order ODE as a system of first order ODEs is to regard the derivatives of the unknown function as additional unknown functions. In your case, regard $f'$ as a new function $g$. Then the system of first order ODEs consists of two equations, the first being the original equation with $f'$ replaced by $g$ and $f''$ replaced by $g'$, and the second equation being $f'=g$.