Clairaut’s equation

The ordinary differential equation

$y=x{\displaystyle \frac{dy}{dx}}+\psi \left({\displaystyle \frac{dy}{dx}}\right),$ (1)

where $\psi$ is a given differentiable real function, is called Clairaut’s equation.

For solving the equation we use an auxiliary variable  $p=:\frac{dy}{dx}$  and write (1) as

$$y=px+\psi (p).$$

Differentiating this equation gives

$$p=x\frac{dp}{dx}+p+{\psi }^{\prime }(p)\frac{dp}{dx},$$

or

$$[x+{\psi }^{\prime }(p)]\frac{dp}{dx}=\mathrm{\hspace{0.33em}0}.$$

The zero rule of product now yields the alternatives

${\displaystyle \frac{dp}{dx}}=\mathrm{\hspace{0.33em}0}$ (2)

and

$x+{\psi }^{\prime }(p)=\mathrm{\hspace{0.33em}0}.$ (3)

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

$y=Cx+\psi (C)$ (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

$y=xp(x)+\psi (p(x)),$ (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

$$y=px+\psi (p),x+{\psi }^{\prime }(p)=\mathrm{\hspace{0.33em}0}.$$

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

Example.  The Clairaut’s equation

$$y=x\frac{dy}{dx}+\frac{a\frac{dy}{dx}}{\sqrt{1+{(\frac{dy}{dx})}^2}}$$

has the general solution

$$y=Cx+\frac{Ca}{\sqrt{1+C^2}}$$

and the singular solution

$$\{\begin{array}{cc}x=-\frac{a}{{(1+p^2)}^{3/2}},\hfill & \\ y=-\frac{a{p}^3}{{(1+p^2)}^{3/2}}\hfill & \end{array}$$

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

$$\sqrt[3]{x^2}+\sqrt[3]{y^2}=\sqrt[3]{a^2},$$

which can be recognized to be the equation of an astroid.  The envelope (see “determining envelope (http://planetmath.org/DeterminingEnvelope)”) of the lines is only the left half of this curve ($x\leqq 0$).  The usual parametric of the astroid is  $x=a{\cos }^3\phi$,  $y=a{\sin }^3\phi$  ().

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