Definitions

# Clairaut's equation

In mathematics, a Clairaut's equation is a differential equation of the form

$y\left(x\right)=xfrac\left\{dy\right\}\left\{dx\right\}+fleft\left(frac\left\{dy\right\}\left\{dx\right\}right\right).$

To solve such an equation, we differentiate with respect to x, yielding

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

so

$0=left\left(x+f\text{'}left\left(frac\left\{dy\right\}\left\{dx\right\}right\right)right\right)frac\left\{d^2 y\right\}\left\{dx^2\right\}.$

Hence, either

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

or

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

In the former case, C = dy/dx for some constant C. Substituting this into the Clairaut's equation, we have the family of functions given by

$y\left(x\right)=Cx+f\left(C\right),,$

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

The latter case,

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

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 represents dy/dx.

This equation was named after Alexis Clairaut.

A first-order partial differential equation is also known as Clairaut's equation or Clairaut equation:

$displaystyle u=xu_x+yu_y+f\left(u_x,u_y\right).$