Clairaut's equation
Type of ordinary differential equation From Wikipedia, the free encyclopedia
Type of ordinary differential equation From Wikipedia, the free encyclopedia
In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form
where 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.[1]
To solve Clairaut's equation, one differentiates with respect to , yielding
so
Hence, either
or
In the former case, for some constant . Substituting this into the Clairaut's equation, one obtains the family of straight line functions given by
the so-called general solution of Clairaut's equation.
The latter case,
defines only one solution , 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 , where .
The parametric description of the singular solution has the form
where is a parameter.
The following curves represent the solutions to two Clairaut's equations:
In each case, the general solutions are depicted in black while the singular solution is in violet.
By extension, a first-order partial differential equation of the form
is also known as Clairaut's equation.[2]
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