Emden’s Equation. A classical nonlinear equation that

Chapter 8, Problem 34E

(choose chapter or problem)

Emden’s Equation. A classical nonlinear equation that occurs in the study of the thermal behavior of a spherical cloud is Emden’s equation

\(y^{\prime \prime}+\frac{2}{x} y^{\prime}+y^{n}=0\)

with initial conditions \(y(0)=1, y^{\prime}(0)=0\). Even though \(x=0 \) is not an ordinary point for this equation (which is nonlinear for \(n \neq 1\)), it turns out that there does exist a solution analytic at \(x=0 \) Assuming that n is a positive integer, show that the first few terms in a power series solution are

\(y=1-\frac{x^{2}}{3 !}+n \frac{x^{2}}{5 !}+\ldots\)

[Hint: Substitute \(y=1+c_{2} x^{2}+c_{3} x^{3}+c_{4} x^{4}+c_{5} x^{5}+\ldots\) into the equation and carefully compute the first few terms in the expansion for \(y^{n}\).]