math lessons - Clairaut's equation

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

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

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

Hence, either

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

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

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

Categories: Ordinary differential equations | Equations

01-04-2007 01:18:14
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