A solution at every point of which the uniqueness of the solution of the
Cauchy problem
for this equation is violated. For example, for an equation of the first order
with a continuous right-hand side which has a finite
or infinite partial derivative everywhere with respect to
,
a singular solution can only lie in the set
A curve
is a singular solution of
(*)
if
is an
integral curve
of the equation
(*)
and if at least one more
integral curve of
(*)
passes through every point of
.
Let equation
(*)
have a
general integral
in a domain
;
if this family of curves has an
envelope,
then this is a singular solution of equation
(*).
For a differential equation
a singular solution is found by examining the
discriminant curve.