Definitions

lexi clairaut

Clairaut's equation

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

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'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 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 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).

External links

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

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