Talk:Lane–Emden equation

I don't understand the word "also" in the first sentence. Michael Hardy 02:04, 16 Oct 2003 (UTC)

Overhaul
I'm currently working on substantially overhauling this article. My work so far is contained in my Userspace at this page: User:Warrickball/Lane–Emden_equation. It's not perfect, and will hopefully improve as others edit it over time. I've tried to add a derivation of the equation and a new figure with the computed solutions. I still intend to add a section about the homology relations and the transformed equation, as well as the isothermal version of the equation. I hope to ultimately merge polytropes. It doesn't make sense to me to discuss the equation and its solutions in separate articles. Warrickball (talk) 17:40, 21 June 2011 (UTC)

I think there's a mistake regarding analytical (exact) solutions. The article says there exist exact solutions only for n = 0, 1, 3 and mentions something about n = 5, but gives solutions only for n = 0, 1, 5. Is there an exact solution for n = 3 then? 89.143.152.219 (talk) 18:31, 11 March 2013 (UTC)

Hi, There isn't an analytical solution for n=3. Also, I added a more detailed explanation of the solution for n=5; I would have added it to your overhaul page but I didn't see your post until just now. TheGhostOfCarlSagan (talk) 17:15, 6 April 2016 (UTC)

n = 0 case
How can n = 0? n is defined at the top in "The index n is the polytropic index that appears in the polytropic equation of state, $$P = K \rho^{1 + \frac{1}{n}}\,$$. So if n = 0, then what equation of state would that be? $$P = K \rho^{\infty}\,$$? --24.130.146.226 (talk) 07:25, 25 October 2013 (UTC)

If one takes the limit $$n\to\infty$$, things remain well-behaved. Physically, it correspond to the case where the density is constant. Warrickball (talk) 12:32, 17 March 2015 (UTC)

n=5 "derivation" makes no sense
The section on the n=5 case *seems* to be presenting how to solve the equation, but the steps make no sense. There's no way to get from the first line (the equation for n=5) to the second line (an expression for the derivative of the solution) without already having solved the equation. The author appears to be demonstrating that the solution works, but not very clearly. Do we need this at all?

The actual solution method was given by Chandrasekhar in a 1967 book; see https://mathworld.wolfram.com/Lane-EmdenDifferentialEquation.html Beanyk (talk) 19:35, 16 August 2023 (UTC)