# Clairaut's equation

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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.

## Solution

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 $p=dy/dx$.
The parametric description of the singular solution has the form
    $x(t)=-f'(t),\,$
    $y(t)=f(t)-tf'(t),\,$
where $t$ is a parameter.

## Examples

The following curves represent the solutions to two Clairaut's equations:
* $f(p)=p^{2}$
* $f(p)=p^{3}$
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
    $\displaystyle u=xu_{x}+yu_{y}+f(u_{x},u_{y})$
is also known as Clairaut's equation.