Emden equation

The non-linear second-order ordinary differential equation

(1)

or, in self-adjoint form,

where , , is a constant. The point is singular for the Emden equation. By the change of variable equation (1) becomes

and by the change of variable ,

After the changes of variables

and subsequent lowering of the order by the substitution , one obtains the first-order equation

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 , , defined on a certain segment , , and having the properties

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

More general than Emden's equation is the Fowler equation

and the Emden–Fowler equation

(2)

where , , are real parameters. As a special case this includes the Thomas–Fermi equation

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

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

(on this and its analogue of order 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)