- Definition
- Examples
- Extension
- See also
- Notes
- References
In mathematics, more specifically in mathematical analysis, Clairaut's equation (or Clairaut equation) is a differential equation of the form
where f is continuously differentiable. It is a particular case of the Lagrange differential equation.
This equation is named after the French mathematician Alexis Clairaut, who introduced it in 1734.[1]
Definition
To solve Clairaut's equation, we differentiate with respect to x, yielding
so
Hence, either
or
In the former case, C = dy/dx for some constant C. Substituting this into the Clairaut's equation, we have the family of straight line functions given by
the so-called general solution of Clairaut's equation.
The latter case,
defines only one solution y(x), the so-called singular solution, whose graph is the envelope of the graphs of the general solutions. The singular solution is usually represented using parametric notation, as (x(p), y(p)), where p = dy/dx.
Examples
The following curves represent the solutions to two Clairaut's equations:
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
is also known as Clairaut's equation.[2]
See also
- Chrystal's equation
Notes
2. ^{{harvnb|Kamke|1944}}.
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
| ref = harv
}}.
- {{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
| ref = harv
}}.
- {{springer
| title = Clairaut equation
| id = C/c022350
| last = Rozov
| first = N. Kh.
}}.
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