# 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

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

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)