# Singular Solution

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## singular solution

[′siŋ·gyə·lər sə′lü·shən]## Singular Solution

A singular solution of a differential equation is a solution at every point of which uniqueness is violated. For the equation y′ = *f(x, y*), this means that through each point of the singular solution there pass several different integral curves with the same tangent. When *f(x, y*) is continuous, this situation is possible only if a Lipschitz condition for *y* is not satisfied at the points of the singular solution for *f(x, y*). For example, the line *y* = *x* is a singular solution for the equation (see Figure 1); through any point *M*_{0} (*x*_{0}, *y*_{0}) of this line, in addition to the line itself, there pass the integral curves

Geometrically, the singular solution is the envelope of the family of integral curves Φ (*x, y, C*) = 0 that form the general integral of the equation.

For the differential equation *F(x, y, y′*) = 0, the discriminant curve *D(x, y*) = 0 is defined as the result of the elimination of the parameter *p = y′* from the system *F(x, y, p*) = 0 and F’_{p} (*x, y, p*) = 0. Only part of this curve is, in general, a singular solution.

### REFERENCE

Stepanov, V. V.*Kurs differentsial’nykh uravnenii*, 8th ed. Moscow, 1959.