# Difference between revisions of "Singular solution"

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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 | ||

− | + | $$ \tag{* } | |

+ | y ^ \prime = f( x, y) | ||

+ | $$ | ||

− | with a continuous right-hand side which has a finite or infinite partial derivative everywhere with respect to | + | 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 | + | A curve $ \gamma \subset M $ |

+ | is a singular solution of (*) if $ \gamma $ | ||

+ | is an [[Integral curve|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|general integral]] $ \phi ( x, y, c) = 0 $ | ||

+ | in a domain $ G $; | ||

+ | if this family of curves has an [[Envelope|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|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> | ||

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====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=14548