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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.

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    Is $f(1)=1$ really what you want? This will give division by zero on the right-hand side! Also, do you really have a boundary value problem (conditions at two different points $R=0$ and $R=1$), or did you mean an initial value problem with two conditions at one point (say $R=0$)? An IVP is relatively straightforward to integrate numerically, but a BVP is harder.2010-10-16
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    I am trying to replicate a graph of the solution from an old paper. I can show a picture of what the graph looks like but I am not sure how. Anyway the solution is for R between 0 and 1. At f(1)=1 at least from what the graphs shows. My other boundary condition is actually wrong (I will edit the main post), but f seems to be approaching positive infinity as R approached 0.2010-10-16
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    Well I am not able to post the graph yet. I don't enough reputation points (i need 10). I might post a link later.2010-10-16
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    More than posting the graph, where did this graph come from, and how did you construct your DE from said graph? (+1 so you can post the graph).2010-10-17
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    This is from an old research paper on Inertial Electrostatic Confinement, in English its a large vacuum tank with hot plasma inside. This equation describes the normalized potential of the plasma at different radii from the center of this spherical tank.2010-10-19
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    Since you talk about spherical tank, I guess you have the homogeneous Neumann boundary condition at 0. It would help if you post the original 3D equation that must hold inside the tank.2010-12-24
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    You might find [this blog post](http://blogs.mathworks.com/loren/2013/06/10/from-symbolic-differential-equations-to-their-numeric-solution/) from The MathWorks helpful.2014-12-01
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    possible duplicate of [How to reduce higher order linear ODE to a system of first order ODE?](http://math.stackexchange.com/questions/501745/how-to-reduce-higher-order-linear-ode-to-a-system-of-first-order-ode) – note that there is no difference between linear and nonlinear in the answer given. This question has been answered many times on this site so I'm sure you can find more examples if you search.2014-12-01

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