Singular Solutions

A singular solution of a first-order ODE is a solution curve that:

• satisfies $F(x,y,y^{\prime })=0)$,
• but is not obtained by any choice of the integration constant $C$,
• instead is the envelope of a family of solutions.

If a family of solutions is

$$\varphi (x,y,C)=0$$

then its envelope is found by solving

$$\frac{\partial \varphi }{\partial C}=0,\varphi (x,y,C)=0$$

and eliminating $C$.

The resulting curve $\psi (x,y)=0$ is the singular solution.

Geometrically: the singular solution is tangent to every nearby member of the family of solutions. This is what makes it another solution.