Clairaut's equation

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In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form

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

where <math>f</math> is continuously differentiable. It is a particular case of the Lagrange differential equation. It is named after the French mathematician Alexis Clairaut, who introduced it in 1734.<ref>Template:Harvnb.</ref>

Solution

To solve Clairaut's equation, one differentiates with respect to <math>x</math>, 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>\frac{d^2 y}{dx^2}\cdot\left[x+f'\left(\frac{dy}{dx}\right)\right] = 0</math>

Hence, either

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

In the former case, <math>\frac{dy}{dx}=C</math> for some constant <math>C</math>. Substituting this into the Clairaut's equation, one obtains the family of straight line functions given by

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

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

The latter case,

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

defines only one solution <math>y(x)</math>, 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 <math>(x(p), y(p))</math>, where <math>p = \frac{dy}{dx}</math>.

The parametric description of the singular solution has the form

<math>x(t)= -f'(t),\,</math>

<math>y(t)= f(t) - tf'(t),\,</math>

where <math>t</math> is a parameter.

Examples

The following curves represent the solutions to two Clairaut's equations:

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In each case, the general solutions are depicted in black while the singular solution is in violet.

Extension

By extension, a first-order partial differential equation of the form

<math>\displaystyle u=xu_x+yu_y+f(u_x,u_y)</math>

is also known as Clairaut's equation.<ref>Template:Harvnb.</ref>

See also

• D'Alembert's equation
• Chrystal's equation
• Legendre transformation

References

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