Singular solution

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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

 (*)

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

A curve is a singular solution of (*) if is an integral curve of the equation (*) and if at least one more integral curve of (*) passes through every point of . Let equation (*) have a general integral in a domain ; if this family of curves has an envelope, then this is a singular solution of equation (*). For a differential equation

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)