View source for Clairaut's equation

{{Short description|Type of ordinary differential equation}}
{{other uses|Clairaut's formula (disambiguation)}}
{{Differential equations}}

In [[mathematical analysis]], '''Clairaut's equation''' (or the '''Clairaut equation''') is a [[differential equation]] of the form

:<math>y(x)=x\frac{dy}{dx}+f\left(\frac{dy}{dx}\right)</math>

where <math>f</math> is [[Differentiable function#Differentiability classes|continuously differentiable]]. It is a particular case of the Lagrange differential equation. It is named after the French [[mathematician]] [[Alexis Clairaut]], who introduced it in 1734.<ref>{{harvnb|Clairaut|1734}}.</ref>

== Solution ==

To solve Clairaut's equation, one differentiates with respect to <math>x</math>, yielding

:<math>\frac{dy}{dx}=\frac{dy}{dx}+x\frac{d^2 y}{dx^2}+f'\left(\frac{dy}{dx}\right)\frac{d^2 y}{dx^2},</math>

so

:<math>\left[x+f'\left(\frac{dy}{dx}\right)\right]\frac{d^2 y}{dx^2} = 0.</math>

Hence, either

:<math>\frac{d^2 y}{dx^2} = 0</math>

or

:<math>x+f'\left(\frac{dy}{dx}\right) = 0.</math>

In the former case, <math>C = dy/dx</math> for some constant <math>C</math>. Substituting this into the Clairaut's equation, one obtains the family of straight line functions given by

:<math>y(x)=Cx+f(C),\,</math>

the so-called ''general solution'' of Clairaut's equation.

The latter case,

:<math>x+f'\left(\frac{dy}{dx}\right) = 0,</math>

defines only one solution <math>y(x)</math>, the so-called ''[[singular solution]]'', whose graph is the [[envelope (mathematics)|envelope]] of the graphs of the general solutions. The singular solution is usually represented using parametric notation, as <math>(x(p), y(p))</math>, where <math>p = dy/dx</math>.

The parametric description of the singular solution has the form

:<math>x(t)= -f'(t),\,</math>
:<math>y(t)= f(t) - tf'(t),\,</math>

where <math>t</math> is a parameter.

== Examples ==

The following curves represent the solutions to two Clairaut's equations:

<gallery>
Image:Solutions to Clairaut's equation where f(t)=t^2.png|{{center|1=<math>f(p)=p^2</math>}}
Image:Solutions to Clairaut's equation where f(t)=t^3.png|{{center|1=<math>f(p)=p^3</math>}}
</gallery>

In each case, the general solutions are depicted in black while the singular solution is in violet.

== Extension ==

By extension, a first-order [[partial differential equation]] of the form
 
:<math>\displaystyle u=xu_x+yu_y+f(u_x,u_y)</math>

is also known as Clairaut's equation.<ref>{{harvnb|Kamke|1944}}.</ref>

==See also==
{{Portal|Mathematics}}
*[[D'Alembert's equation]]
*[[Chrystal's equation]]
*[[Legendre transformation]]

== Notes ==
{{Reflist}}

== References ==

*{{Citation
 | last = Clairaut
 | first = Alexis Claude
 | title = Solution de plusieurs problèmes où il s'agit de trouver des Courbes dont la propriété consiste dans une certaine relation entre leurs branches, exprimée par une Équation donnée.
 | url = http://gallica.bnf.fr/ark:/12148/bpt6k3531x/f344.table
 | journal = Histoire de l'Académie Royale des Sciences
 | year = 1734
 | pages = 196–215
 }}.

*{{Citation
 | last = Kamke
 | first = E.
 | language = de
 | title = Differentialgleichungen: Lösungen und Lösungsmethoden
 | volume = 2. Partielle Differentialgleichungen 1er Ordnung für eine gesuchte Funktion
 | publisher = Akad. Verlagsgesell
 | year = 1944
 }}.

*{{springer
 | title = Clairaut equation
 | id = C/c022350
 | last = Rozov
 | first = N. Kh.
}}.

[[Category:Ordinary differential equations]]