Clairaut's equation

From Academic Kids

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

<math>y(x)=x\frac{dy}{dx}+f\left(\frac{dy}{dx}\right).<math>

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

<math>\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},<math>

so

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

Hence, either

<math>0=\frac{d^2 y}{dx^2}<math>

or

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

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

<math>y(x)=Cx+f(C),\,<math>

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

The latter case,

<math>0=x+f'\left(\frac{dy}{dx}\right),<math>

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.de:Clairaut-Gleichung it:equazione di Clairaut

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