Singular Solution - Failure of Uniqueness

Failure of Uniqueness

Consider the differential equation

A one-parameter family of solutions to this equation is given by

Another solution is given by

Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. By considering the two sets of solutions above, one can see that the solution fails to be unique when . (It can be shown that for, if a single branch of the square root is chosen, then there is a local solution which is unique using the Picard–Lindelöf theorem.) Thus, the solutions above are all singular solutions, in the sense that solution fails to be unique in a neighbourhood of one or more points. (Commonly, we say "uniqueness fails" at these points.) For the first set of solutions, uniqueness fails at one point, and for the second solution, uniqueness fails at every value of . Thus, the solution is a singular solution in the stronger sense that uniqueness fails at every value of x. However, it is not a singular function since it and all its derivatives are continuous.

In this example, the solution is the envelope of the family of solutions . The solution is tangent to every curve at the point .

The failure of uniqueness can be used to construct more solutions. These can be found by taking two constant and defining a solution to be when, to be when, and to be when . Direct calculation shows that this is a solution of the differential equation at every point, including and . Uniqueness fails for these solutions on the interval, and the solutions are singular, in the sense that the second derivative fails to exist, at and .