A solution, , of a differential equation is said to be * stable* if
any other solution whose initial data is sufficiently close to that of
remains in a ``tube'' enclosing ; if the solution is not stable,
it is said to be * unstable*. If the diameter of the tube approaches
zero as **t** becomes large, then is said to be * asymptotically
stable*.

In elementary treatments of differential equations it is assumed that the
initial value problem has a unique solution that exists throughout the
interval of interest and which can be obtained by analytical techniques.
However, many of the differential equations encountered in practice cannot
be solve explicitly, so we are led to methods for obtaining approximations
to solutions. Such solutions are usually called * numerical* solutions.
Matters are also complicated by the fact that solutions can fail to exist
over the desired interval of interest. Even more troublesome are problems
with more than one solution.

a) The differential equation,

does not have a solution that can be expressed in terms of elementary functions.

b) The IVP,

has the solution which exists on the interval but does not exist on the interval .

c) The IVP,

does not have a unique solution. In fact, it is not difficult to show that:

1) is a solution on any interval containing **t = 0**;

2) is a solution on any interval for any ** b > 0**;

3) is a solution on and this is the largest such interval on which is a solution.