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''of an ordinary differential equation''
 
''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
 
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>
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$$ \tag{* }
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y  ^  \prime  = f( x, y)
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$$
  
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
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with a continuous right-hand side which has a finite or infinite partial derivative everywhere with respect to $  y $,  
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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>
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$$
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= \{ {( x, y) } : {| f _ {y} ^ { \prime } ( x, y) | = \infty } \}
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.
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$$
  
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
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A curve $  \gamma \subset  M $
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is a singular solution of (*) if $  \gamma $
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is an [[Integral curve|integral curve]] of the equation (*) and if at least one more integral curve of (*) passes through every point of $  \gamma $.  
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Let equation (*) have a [[General integral|general integral]] $  \phi ( x, y, c) = 0 $
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in a domain $  G $;  
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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>
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$$
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F( x, y, y  ^  \prime  )  = 0 ,
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$$
  
 
a singular solution is found by examining the [[Discriminant curve|discriminant curve]].
 
a singular solution is found by examining the [[Discriminant curve|discriminant curve]].
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====References====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 08:14, 6 June 2020


of an ordinary differential equation

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

$$ \tag{* } y ^ \prime = f( x, y) $$

with a continuous right-hand side which has a finite or infinite partial derivative everywhere with respect to $ y $, a singular solution can only lie in the set

$$ M = \{ {( x, y) } : {| f _ {y} ^ { \prime } ( x, y) | = \infty } \} . $$

A curve $ \gamma \subset M $ is a singular solution of (*) if $ \gamma $ is an integral curve of the equation (*) and if at least one more integral curve of (*) passes through every point of $ \gamma $. Let equation (*) have a general integral $ \phi ( x, y, c) = 0 $ in a domain $ G $; if this family of curves has an envelope, then this is a singular solution of equation (*). For a differential equation

$$ F( x, y, y ^ \prime ) = 0 , $$

a singular solution is found by examining the discriminant curve.

References

[1] W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[2] G. Sansone, "Ordinary differential equations" , 2 , Zanichelli (1948) (In Italian)

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. [a1].

References

[a1] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) pp. §§3.6, 3.51, 4.7, A.5
How to Cite This Entry:
Singular solution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Singular_solution&oldid=48722
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article