How difficult is it to solve this elliptic ODE.

[ozone]

Hello, I am working on a research problem and I am not sure whether or not I will be able to figure this out in a suitable amount of time. I have never solved a single elliptic integral and they do seem non-trivial to gain an understanding of (most of the books I've glanced at assume a very high level knowledge of differential equations). I do not need an exact solution to this integral(I am in the regime where 1/r^3 <<1), but I am just curious how difficult it would be to obtain an approximate solution to r(λ) for

(\frac{dr}{dλ})^2 = E^2 - \frac{L^2}{2r^2} + \frac{GML^2}{r^3}

This is an important result for physics, namely it is a radial null geodesic in the Schwarzschild metric. Usually substitutions are made to bring it to a differential equation of dr/d(phi), but I want to know the solution in terms of the affine parameter λ.

I tried separating, solving the radial coordinate and then back-solving for λ, but this was very far from being useful. Mathematica is not much help in solving this ode sadly, but I do recognize that it should have a solution in terms of elliptic functions. I even tried a u=1/r substitution and did not gain any ground that way. Any thoughts or tips on good resources and how difficult this should be to solve are welcomed.

 

[ShayanJ]

Try www.wolframalpha.com , but don't expect a pretty answer!

 

[JJacquelin]

Hi !
it's a separable ODE. So the function λ(r) can be expressed on integral form (attachment)
Then, an approximate is obtainded on the form of a limited series expansion. For less deviation, expand it with more terms.
If you need an approximate of the function r(λ), you have to inverse the series.