Clairaut's equation

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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},

so

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}

or

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 straight line 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.

This equation has been named after Alexis Clairaut, who has introduced it in 1734.

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).

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