Singular Solution — Worked Example — First-Order ODE

A singular solution of a differential equation is a solution that is not obtained by substituting a constant value into the parameter of the general solution.

Example

Given the differential equation

\( y - x y' - 2 (y')^2 = 0, \)

we compute a general solution. Denote \( p = y' \) and rearrange:

\( y = xp + 2p^2. \)

Differentiating with respect to \( x \) gives

\( (x + 4p)\, \frac{dp}{dx} = 0. \)

If \( \dfrac{dp}{dx} = 0 \), then \( p = a \) is constant, and we obtain the general solution:

\( y = ax + 2a^2. \)

If \( x + 4p = 0 \), then \( p = -\dfrac{x}{4} \). Substituting into \( y = xp + 2p^2 \):

\( y = x\!\left( -\frac{x}{4} \right) + 2\!\left( -\frac{x}{4} \right)^{\!2} = -\frac{x^2}{4} + \frac{x^2}{8} = -\frac{x^2}{8}, \)

and we obtain the singular solution:

\( x^2 + 8y = 0. \)

The line family \( y = ax + 2a^2 \) sweeps over \( a \), and the singular solution — the parabola \( x^2+8y=0 \) — emerges as its envelope

Summary

The general solution is the family of lines \( y = ax + 2a^2 \), and the singular solution is the parabola \( x^2 + 8y = 0 \), which is an envelope of the general solution: every line of the family is tangent to it at some point.

In other words: a singular solution of a differential equation is a solution that is not obtained by substituting a constant value into the parameter of the general solution, yet every general solution is tangent to it at some point. That is, the singular solution is the envelope of the graphs of the general solution.