An ordinary first-order differential equation not
solved with respect to its derivative:
where
is a non-linear function. Equation
(1)
is named after
A. Clairaut
[1]
who was the first to point out the difference
between the general and the singular solutions of an equation of
this form. The Clairaut equation is a particular case of the
Lagrange equation.
If
and
when
,
then the set of integral curves (cf.
Integral curve)
of
(1)
consists of: a parametrically given curve
a one-parameter family of straight lines
tangent to the curve
(2);
curves consisting of an arbitrary segment of
the curve
(2)
and the two straight lines of the family
(3)
tangent
to
(2)
at each end of this segment. The family
(3)
forms the
general solution,
while the curve
(2),
which is the
envelope
of the family
(3),
is the
singular solution
(see
[2]).
A family of tangents to a smooth non-linear curve
satisfies a Clairaut equation. Therefore, geometric problems in which it is
required to determine a curve in terms of a prescribed property
of its tangents (common to all points of the curve) leads to a Clairaut equation.
The following first-order
partial differential equation
is also called a Clairaut equation:
it has the integral
where
is an arbitrary point of the domain of definition of the function
(see
[3]).