Definition:Singular Solution to Differential Equation

Definition

A singular solution to a differential equation $E$ is a general solution to $E$ which does not contain an arbitrary constant, and hence forms a particular solution on its own.

Also known as

Some sources refer to a singular solution as a singular integral.

Also see

• Definition:Solution of Differential Equation
• Definition:Particular Solution to Differential Equation
• Definition:General Solution to Differential Equation

Historical Note

Singular solutions were noted by Gottfried Wilhelm von Leibniz in $1694$, and also by Brook Taylor in $1715$.

However, they are generally associated with Alexis Claude Clairaut, who reported on them in $1734$.

Sources

• 1956: E.L. Ince: Integration of Ordinary Differential Equations (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: Equations of the First Order and Degree: $2$. Integration