# Clairaut’s equation

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

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

The zero rule of product now yields the alternatives

 $\displaystyle\frac{dp}{dx}\;=\;0$ (2)

and

 $\displaystyle x+\psi^{\prime}(p)\;=\;0.$ (3)

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

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

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

 $y\;=\;px+\psi(p),\qquad x+\psi^{\prime}(p)\;=\;0.$

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

 $\begin{cases}x\;=\;-\frac{a}{(1\!+\!p^{2})^{3/2}},\\ y\;=\;-\frac{ap^{3}}{(1+p^{2})^{3/2}}\\ \end{cases}$

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  $x=a\cos^{3}\varphi$,  $y=a\sin^{3}\varphi$  ($0\leqq\varphi<2\pi$).

## References

• 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