Emden equation

The non-linear second-order ordinary differential equation

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

or, in self-adjoint form,

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

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]).

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