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

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

To solve such an equation, we differentiate 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

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

Hence, either

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

or

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

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

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

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

The latter case,

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

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 has been named after Alexis Clairaut, who has introduced it in 1734.

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

$\displaystyle u=xu_{x}+yu_{y}+f(u_{x},u_{y}).$

Examples

• 

  Solutions to Clairaut's equation where $f(p)=p^{2}$

• 

  $f(p)=p^{3}$

External links

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