Clairaut's equation
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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