Clairaut's equation
Template:Short description {{#invoke:other uses|otheruses}} Template:Differential equations
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>
SolutionEdit
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>\left[x+f'\left(\frac{dy}{dx}\right)\right]\frac{d^2 y}{dx^2} = 0.</math>
Hence, either
- <math>\frac{d^2 y}{dx^2} = 0</math>
or
- <math>x+f'\left(\frac{dy}{dx}\right) = 0.</math>
In the former case, <math>C = dy/dx</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>x+f'\left(\frac{dy}{dx}\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 = 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.
ExamplesEdit
The following curves represent the solutions to two Clairaut's equations:
- Solutions to Clairaut's equation where f(t)=t^2.png
style }}
- Solutions to Clairaut's equation where f(t)=t^3.png
style }}
In each case, the general solutions are depicted in black while the singular solution is in violet.
ExtensionEdit
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>