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[parent] Clairaut's equation (Derivation)

The ordinary differential 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

$\displaystyle y = px+\psi(p).$
Differentiating this equation gives
$\displaystyle p = x\frac{dp}{dx}+p+\psi'(p)\frac{dp}{dx}$
or
$\displaystyle [x+\psi'(p)]\frac{dp}{dx} = 0.$
The zero rule of product now yields the alternatives
$\displaystyle \frac{dp}{dx} = 0$ (2)

and
$\displaystyle x+\psi'(p) = 0.$ (3)

Integrating (2) we get $ p = C$ (constant), 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 terms 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
$\displaystyle y = px+\psi(p), x+\psi'(p) = 0.$
Thus the singular solution presents the envelope of the family (4).

Example. The Clairaut's equation

$\displaystyle y = x\frac{dy}{dx}+\frac{a\frac{dy}{dx}}{\sqrt{1+(\frac{dy}{dx})^2}}$
has the general solution
$\displaystyle y = Cx+\frac{Ca}{\sqrt{1+C^2}}$
and the singular solution
\begin{displaymath}\begin{cases} x = -\frac{a}{(1+p^2)^{3/2}},\ y = -\frac{ap^3}{(1+p^2)^{3/2}}\ \end{cases}\end{displaymath}
in a parametric form. Eliminating the parameter $ p$ yields the form
$\displaystyle \sqrt[3]{x^2}+\sqrt[3]{y^2} = \sqrt[3]{a^2},$
which can be recognized to be the equation of an astroid. The envelope (see “determining envelope”) of the lines is only the left half of this curve ($ x \leqq 0$). The usual parametric presentation of the astroid is $ x = a\cos^3\varphi$, $ y = a\sin^3\varphi$ ( $ 0 \leqq \varphi < 2\pi$).
\includegraphics{clairaut2}

Bibliography

1
N. PISKUNOV: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele. - Kirjastus Valgus, Tallinn (1966).



"Clairaut's equation" is owned by pahio. [ full author list (2) ]
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See Also: d'Alembert's equation, famous curves, index of differential equations, perimeter of astroid, singular solution

Other names:  Clairaut differential equation
Also defines:  astroid

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Cross-references: curve, parametric form, envelope, parameter, singular solution, solution, easy to see, lines, straight, general solution, zero rule of product, auxiliary variable, equation, real function, differentiable, ordinary differential equation
There are 5 references to this entry.

This is version 16 of Clairaut's equation, born on 2005-05-07, modified 2007-06-02.
Object id is 7024, canonical name is ClairautsEquation.
Accessed 6998 times total.

Classification:
AMS MSC: 34C05 (Ordinary differential equations :: Qualitative theory :: Location of integral curves, singular points, limit cycles)
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