# Emden equation

The non-linear second-order ordinary differential equation

$$\tag{1 } \frac{d ^ {2} y }{d x ^ {2} } + \frac{2}{x} \frac{d y }{d x } + y ^ \alpha = 0 ,$$

$$\frac{d}{dx} \left ( x ^ {2} \frac{d y }{d x } \right ) + x ^ {2} y ^ \alpha = 0 ,$$

where $\alpha > 0$, $\alpha \neq 1$, is a constant. The point $x = 0$ is singular for the Emden equation. By the change of variable $x = 1 / \xi$ equation (1) becomes

$$\frac{d ^ {2} y }{d \xi ^ {2} } + \frac{y ^ \alpha }{\xi ^ {4} } = 0 ;$$

and by the change of variable $y = \eta / x$,

$$\frac{d ^ {2} \eta }{d x ^ {2} } + \frac{\eta ^ \alpha }{x ^ {\alpha - 1 } } = 0 .$$

After the changes of variables

$$x = e ^ {-} t ,\ y = e ^ {\mu t } u ,\ \ \mu = \frac{2}{( \alpha - 1 ) } ,$$

and subsequent lowering of the order by the substitution $u ^ \prime = v ( u )$, one obtains the first-order equation

$$\frac{d v }{d u } = - ( 2 \mu - 1 ) - \frac{\mu ( \mu - 1 ) \mu + \mu ^ \alpha }{v} .$$

Equation (1) was obtained by R. Emden [1] in connection with a study of equilibrium conditions for a polytropic gas ball; this study led him to the problem of the existence of a solution of (1) with the initial conditions $y ( 0) = 1$, $y ^ \prime ( 0) = 0$, defined on a certain segment $[ 0 , x _ {0} ]$, $0 < x _ {0} < \infty$, and having the properties

$$y ( x) > 0 \ \textrm{ for } 0 \leq x < x _ {0} ,\ \ y ( x _ {0} ) = 0 .$$

Occasionally (1) is also called the Lienard–Emden equation.

More general than Emden's equation is the Fowler equation

$$\frac{d}{dx} \left ( x ^ {2} \frac{dy}{dx} \right ) + x ^ \lambda y ^ \alpha = 0 ,\ \lambda , \alpha > 0 ,$$

and the Emden–Fowler equation

$$\tag{2 } \frac{d}{dx} \left ( x ^ \rho \frac{dy}{dx} \right ) \pm x ^ \lambda y ^ \alpha = 0 ,$$

where $\rho$, $\lambda$, $\alpha \neq 1$ are real parameters. As a special case this includes the Thomas–Fermi equation

$$\frac{d ^ {2} y }{d x ^ {2} } = \ \frac{y ^ {3/2} }{\sqrt x } ,$$

which arises in the study of the distribution of electrons in an atom. If $\rho \neq 1$, then by a change of variables (2) can be brought to the form

$$\frac{d ^ {2} w }{d s ^ {2} } \pm s ^ \sigma w ^ \alpha = 0 .$$

There are various results in the qualitative and asymptotic investigation of solutions of the Emden–Fowler equation (see, for example, [2], [3]). A detailed study has also been made of the equation of Emden–Fowler type

$$\frac{d ^ {2} y }{d x ^ {2} } + a ( x) | y | ^ \alpha \mathop{\rm sign} y = 0$$

(on this and its analogue of order $n$ see [4]).

#### References

 [1] R. Emden, "Gaskugeln" , Teubner (1907) [2] G. Sansone, "Equazioni differenziali nel campo reale" , 2 , Zanichelli (1949) [3] R.E. Bellman, "Stability theory of differential equations" , McGraw-Hill (1953) [4] I.T. Kiguradze, "Some singular boundary value problems for ordinary differential equations" , Tbilisi (1975) (In Russian)
How to Cite This Entry:
Emden equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Emden_equation&oldid=46818
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article