The Lane-Emden Equation describes the relationship between density and radius of a self-gravitating ball of gas (namely, a star). On the next page, the Lane-Emden is Equation 1. To solve this equation, though, we need to know the relationship between pressure and density for the type of gas in the star, which is called the equation of state. The L-E equation assumes that the equation of state takes the form P = Krg, where P is pressure, r is density, and K is some constant. There are a few values of g for which the equation can be solved analytically – these are not relevant to physical stars, but they are useful as a check for your R-K code. These are for n=0, n=1, and n=5, where n=1/(g-1), and the solutions are given below.
You should do parts a and b of the problem below. You start at the center, where a=0 and x=1 (d x /da=0 at the center, to ensure continuous density). Integrate out until x drops to zero, then you know you are at the surface of the star, and the value of a at that point tells you the radius. Once you know those numbers, Equations 2 and 4 allow you to calculate the central pressure and density.
You’ll probably want to talk to me about the details.