View source for Singular solution

''of an ordinary differential equation''

A solution at every point of which the uniqueness of the solution of the [[Cauchy problem|Cauchy problem]] for this equation is violated. For example, for an equation of the first order

<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/s/s085/s085610/s0856101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

with a continuous right-hand side which has a finite or infinite partial derivative everywhere with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085610/s0856102.png" />, a singular solution can only lie in the set

<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/s/s085/s085610/s0856103.png" /></td> </tr></table>

A curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085610/s0856104.png" /> is a singular solution of (*) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085610/s0856105.png" /> is an [[Integral curve|integral curve]] of the equation (*) and if at least one more integral curve of (*) passes through every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085610/s0856106.png" />. Let equation (*) have a [[General integral|general integral]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085610/s0856107.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085610/s0856108.png" />; if this family of curves has an [[Envelope|envelope]], then this is a singular solution of equation (*). For a 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/s/s085/s085610/s0856109.png" /></td> </tr></table>

a singular solution is found by examining the [[Discriminant curve|discriminant curve]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.W. [V.V. Stepanov] Stepanow,   "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Sansone,   "Ordinary differential equations" , '''2''' , Zanichelli  (1948)  (In Italian)</TD></TR></table>



====Comments====
Under  "singular solution of a differential equation"  is also understood a particular solution that is not obtainable by specifying the integration constant in a general solution. The two notions have much to do with one another but are not identical, cf. [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)  pp. §§3.6, 3.51, 4.7, A.5</TD></TR></table>