Clairaut’s equation

The ordinary differential equationMathworldPlanetmath

y=xdydx+ψ(dydx), (1)

where ψ is a given differentiableMathworldPlanetmathPlanetmath real function, is called Clairaut’s equation.

For solving the equation we use an auxiliary variable  p=:dydx  and write (1) as


Differentiating this equation gives



[x+ψ(p)]dpdx= 0.

The zero rule of product now yields the alternatives

dpdx= 0 (2)


x+ψ(p)= 0. (3)

Integrating (2) we get  p=C (), and substituting this in (1) gives the general solution

y=Cx+ψ(C) (4)

which presents a family of straight lines.

If (3) allows to solve p in of x,  p=p(x),  we can write (1) as

y=xp(x)+ψ(p(x)), (5)

which is easy to see satisfying (1).  The solution (5) may not be gotten from (4) using any value of C.  It is a singular solution which may be obtained by eliminating the parameter p from the equations

y=px+ψ(p),x+ψ(p)= 0.

Thus the singular solution presents the envelope of the family (4).

Example.  The Clairaut’s equation


has the general solution


and the singular solution


in a parametric form.  Eliminating the parametre p yields the form


which can be recognized to be the equation of an astroid.  The envelope (see “determining envelope (”) of the lines is only the left half of this curve (x0).  The usual parametric of the astroid is  x=acos3φ,  y=asin3φ  (0φ<2π).


  • 1 N. Piskunov: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele.  – Kirjastus Valgus, Tallinn (1966).
Title Clairaut’s equation
Canonical name ClairautsEquation
Date of creation 2015-02-04 11:20:02
Last modified on 2015-02-04 11:20:02
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 21
Author pahio (2872)
Entry type Derivation
Classification msc 34C05
Synonym Clairaut differential equation
Related topic DAlembertsEquation
Related topic FamousCurvesInThePlane
Related topic IndexOfDifferentialEquations
Related topic PerimeterOfAstroid
Related topic SingularSolution
Related topic DerivativeAsParameterForSolvingDifferentialEquations
Defines astroid