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]).
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) |
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