View source for Emden equation

The non-linear second-order ordinary differential equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e0355601.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

or, in self-adjoint form,

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e0355603.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e0355604.png" />, is a constant. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e0355605.png" /> is singular for the Emden equation. By the change of variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e0355606.png" /> equation (1) becomes

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and by the change of variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e0355608.png" />,

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After the changes of variables

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and subsequent lowering of the order by the substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e03556011.png" />, one obtains the first-order equation

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Equation (1) was obtained by R. Emden [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e03556013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e03556014.png" />, defined on a certain segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e03556015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e03556016.png" />, 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e03556019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e03556020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e03556021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e03556022.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e03556024.png" />, 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, [[#References|[2]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035560/e03556027.png" /> see [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Emden,   "Gaskugeln" , Teubner  (1907)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Sansone,   "Equazioni differenziali nel campo reale" , '''2''' , Zanichelli  (1949)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.E. Bellman,   "Stability theory of differential equations" , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.T. Kiguradze,   "Some singular boundary value problems for ordinary differential equations" , Tbilisi  (1975)  (In Russian)</TD></TR></table>