Singular Solutions — Definitions

General solution
A parametric family of solutions \( y = \varphi(x, C) \) depending on an arbitrary constant \( C \). Every choice of a specific value for \( C \) gives a particular solution.
Singular solution
A solution of the differential equation that is not obtained from the general solution \( y = \varphi(x, C) \) for any value of the constant \( C \).

Types of singular solutions

Envelope
An important special case of a singular solution — a curve that every solution from the general solution is tangent to at some point. The envelope is obtained by eliminating the parameter \( C \) from the system
\( \begin{cases} F(x, y, C) = 0 \\[2pt] \dfrac{\partial F}{\partial C}(x, y, C) = 0 \end{cases} \)
Singular solution that is not an envelope
A solution that is not obtained from the general solution, but is also not tangent to every general solution. A typical example: \( y \equiv 0 \) in Bernoulli equations \( y' + P(x)\,y = Q(x)\,y^n \), which is lost at the division step.

Classification table

Solution typeDefinitionExample
Particular solution Obtained from the general solution by substituting a specific value of \( C \) \( y = 2x + 8 \) from \( y = ax + 2a^2 \) with \( a = 2 \)
Singular solution — envelope Not obtained for any \( C \), and every general solution is tangent to it \( x^2 + 8y = 0 \) for the equation \( y - x y' - 2 (y')^2 = 0 \)
Singular solution — not an envelope Not obtained for any \( C \), but not necessarily tangent to the general solutions \( y \equiv 0 \) in the Bernoulli equation \( y' + P(x)\,y = Q(x)\,y^n \)

Example — an envelope

The line family \( y = ax + 2a^2 \) and the parabola \( x^2 + 8y = 0 \): every line touches the parabola at one point

Consider the equation

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

Its general solution is the family of lines

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

The singular solution is the parabola

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

which is an envelope — every line of the family is tangent to it at exactly one point.

Example — a singular solution that is not an envelope

Consider the equation

\( y' = y^{2/3}. \)

The general solution is

\( y = \frac{(x - C)^3}{27}. \)

The solution \( y \equiv 0 \) is a singular solution: it is not obtained for any value of \( C \), yet it is not an envelope of the family.

The cubics \( y = (x-C)^3/27 \) and the singular solution \( y \equiv 0 \), which is not an envelope
© Shir Sivroni