2nd order DE for planet's density

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SUMMARY

The discussion focuses on solving the second-order differential equation related to the density of a planet as a function of radius using the Lane-Emden equation. The equation presented is θ'' + (2/x)θ' + 1 = 0, where θ represents the density. The solution requires numerical methods implemented in Mathematica, with an emphasis on iteratively adjusting initial conditions to achieve the desired density profile, ensuring maximum density at the center (r = 0) and minimum density at the surface (r = R).

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Homework Statement



I attached the problem because it's easier

Homework Equations





The Attempt at a Solution



The main problem I have with this problem is trying to find the density as a function of radius.
I have been thinking for hours but can't come up with anything.

What I have for the n=0 question:
for clarity, I'm just using x

θ''+(2/x)θ' +1 = 0

is my equation and I'm using mathematica to numerically solve it.
 

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The variable theta is defined as the density of the body as a function of radius. Theta is not given explicitly in terms of r, but the values of theta are determined by solving the Lane-Emden ODE subject to the initial conditions specified in the text. As the text describes, the initial conditions themselves are not explicit, and the Lane-Emden equation must be solved in an iterative fashion while manipulating the initial conditions. Basically, follow the recipe given in the text and check the values of theta to see if they give a maximum value when r = 0 and a minimum value when r = R (the surface of the body).
 

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