Emden equation
The non-linear second-order ordinary differential equation
![]() | (1) |
or, in self-adjoint form,
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where ,
, is a constant. The point
is singular for the Emden equation. By the change of variable
equation (1) becomes
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and by the change of variable ,
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After the changes of variables
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and subsequent lowering of the order by the substitution , one obtains the first-order equation
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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
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Occasionally (1) is also called the Lienard–Emden equation.
More general than Emden's equation is the Fowler equation
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and the Emden–Fowler equation
![]() | (2) |
where ,
,
are real parameters. As a special case this includes the Thomas–Fermi equation
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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
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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
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(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) |
Emden equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Emden_equation&oldid=46818