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1.1 Some Elements of the Theory     continued...

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.