# Clairaut's equation

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

$y(x)=x{\frac {dy}{dx}}+f\left({\frac {dy}{dx}}\right)$ where f is 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.

## Definition

To solve Clairaut's equation, one differentiates with respect to x, yielding

${\frac {dy}{dx}}={\frac {dy}{dx}}+x{\frac {d^{2}y}{dx^{2}}}+f'\left({\frac {dy}{dx}}\right){\frac {d^{2}y}{dx^{2}}},$ so

$\left[x+f'\left({\frac {dy}{dx}}\right)\right]{\frac {d^{2}y}{dx^{2}}}=0.$ Hence, either

${\frac {d^{2}y}{dx^{2}}}=0$ or

$x+f'\left({\frac {dy}{dx}}\right)=0.$ In the former case, C = dy/dx for some constant C. Substituting this into the Clairaut's equation, one obtains the family of straight line functions given by

$y(x)=Cx+f(C),\,$ the so-called general solution of Clairaut's equation.

The latter case,

$x+f'\left({\frac {dy}{dx}}\right)=0,$ 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

$\displaystyle u=xu_{x}+yu_{y}+f(u_{x},u_{y})$ is also known as Clairaut's equation.