Clairaut's equation

For other uses, see Clairaut's formula (disambiguation).

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

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

where ${\displaystyle f}$ 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.^[1]

Solution

To solve Clairaut's equation, one differentiates with respect to ${\displaystyle x}$, yielding

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

so

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

Hence, either

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

or

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

In the former case, ${\displaystyle C=dy/dx}$ for some constant ${\displaystyle C}$. Substituting this into the Clairaut's equation, one obtains the family of straight line functions given by

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

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

The latter case,

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

defines only one solution ${\displaystyle y(x)}$, 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 ${\displaystyle (x(p),y(p))}$, where ${\displaystyle p=dy/dx}$.

The parametric description of the singular solution has the form

${\displaystyle x(t)=-f'(t),\,}$

${\displaystyle y(t)=f(t)-tf'(t),\,}$

where ${\displaystyle t}$ is a parameter.

Examples

The following curves represent the solutions to two Clairaut's equations:

• 
  ${\displaystyle f(p)=p^{2}}$
• 
  ${\displaystyle f(p)=p^{3}}$

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

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

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

See also

• Mathematics portal

• D'Alembert's equation
• Chrystal's equation
• Legendre transformation