Clairaut's equation

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

:$y(x)=x\frac+f\left(\frac\right)$

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

:$\frac=\frac+x\frac+f'\left(\frac\right)\frac,$

so

:\left +f'\left(\frac\right)\rightfrac = 0.

Hence, either

:$\frac = 0$

or

:$x+f'\left(\frac\right) = 0.$

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,

:$x+f'\left(\frac\right) = 0,$

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:

Image:Solutions to Clairaut's equation where f(t)=t^2.png,
Image:Solutions to Clairaut's equation where f(t)=t^3.png,

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..

See also

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

Notes

References

*.

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*{{springer
, title = Clairaut equation
, id = C/c022350
, last = Rozov
, first = N. Kh.
.

Eponymous equations of mathematics
Ordinary differential equations