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

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

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

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

so

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

Hence, either

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

or

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

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

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

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

The latter case,

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

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 was named after Alexis Clairaut.

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

- $displaystyle\; u=xu\_x+yu\_y+f(u\_x,u\_y).$

- . At Gallica: the paper of Clairaut introducing the equation named after him.

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Last updated on Wednesday September 03, 2008 at 21:02:21 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday September 03, 2008 at 21:02:21 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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