singular solution

Let the general solution y of the differential equationMathworldPlanetmath

F(x,y,dxdy)=0 (1)

be given by the equation

G(x,y,C)=0, (2)

the so-called general integral of (1); herein C means an arbitrary .

Suppose that the family (2) of the integral curves of (1) has an envelope.  In any point of the envelope, the tangent line is also the tangent of one integral curve.  Thus in such a point, all the three values x,y,dydx are same for the envelope curve and the integral curve.  But these values satisfy the equation (1).  Accordingly, this equation is satisfied also by the abscissa, ordinate and slope of the envelope.  This means that the envelope is an integral curve of the differential equation (1).  Because the envelope does not belong to the family (2), it cannot be obtained from the equation (2) with any value of C.

A solution of the differential equation (1) which cannot be obtained from the general integral, is called a singular solution of equation.

An envelope of the integral curves means always a singular solution, but sometimes a singular solution is only the locus of the singular pointsMathworldPlanetmathPlanetmathPlanetmath of the curves (e.g. for the equation  3yy=1).

Note.  If the family (2) has an envelope, it may be found by eliminating C from the pair


Example.  Find the singular solution of the equation

y2(1+y)2=R2. (3)

We can separate the variables, getting


Integrating this yields


which represents the family of circles with radius R and centres on the x-axis.  The family has as envelope the double line  y=±R;  the functions  y±R  satisfy also the differental equation (3) and thus are singular solutions of it.

Title singular solution
Canonical name SingularSolution
Date of creation 2013-03-22 18:05:58
Last modified on 2013-03-22 18:05:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Definition
Classification msc 34A05
Synonym singular solution of differential equation
Related topic DeterminingEnvelope
Related topic ClairautsEquation
Related topic SeparationOfVariables
Defines general integral