Clairaut's equation

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Not to be confused with Clairaut's theorem or Clairaut's relation.

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[edit]

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[edit]

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[edit]

By extension, a first-order partial differential equation of the form

is also known as Clairaut's equation.[2]

Notes[edit]

References[edit]

  • Kamke, E. (1944), Differentialgleichungen: Lösungen und Lösungsmethoden (in German), 2. Partielle Differentialgleichungen 1er Ordnung für eine gesuchte Funktion, Akad. Verlagsgesell .