An introduction to some ordinary differential equations governing stellar structures

Sammanfattning: The Lane-Emden equation is a non-linear differential equation governing the equilibrium of polytropic stationary self-gravitating, spherically symmetric star models; $${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left({\xi ^{2}{\frac {d\theta }{d\xi }}}\right)+\theta ^{n}=0.$$ In the isothermal cases we have the Chandrasekhar equation: $${\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left({\xi ^{2}{\frac {d\psi}{d\xi }}}\right)-e^{-\psi}=0$$ After having derived these models, we will go through all cases for which analytic solutions are achievable. Moreover, we will discuss the existence and uniqueness of positive solutions under specific boundary conditions by transforming the equations to autonomous ones. The analysis depends upon the value of the polytropic index $n.$ We also compute some solutions numerically.