cauchy initial value problem

Let be a subset of
, a point of , and
be a function.

We say that a function is a solution to the Cauchy (or initial value) problem

(1)

if

1. is a differentiable function
   defined on a interval
   ;
2. one has
   for all and ;
3. one has
   and
   for all .

We say that a solution
is a maximal solution if it cannot be extended to a bigger interval. More precisely given any other solution
defined on an interval
and such that for all , one has (and hence and are the same function).

We say that a solution
is a global solution if
.

We say that a solution
is unique if given any other solution
one has for all (i.e. is the unique solution defined on the interval ).

Notation

Usually the differential equation in (1) is simply written as . Also, depending on the topics, the name chosen for the function and for the variable, can change. Other common choices are or . It is also common to write
when the independent variable represents a time value.

Examples

1. The function
   defined on
   is the unique maximal solution to the Cauchy problem:

   In this case
   ,
   , , .

2. The function is a global (and hence maximal), unique solution to the Cauchy problem:

3. Consider the Cauchy problem

   The function defined on
   is a global solution. However the function
   defined on
   is also a solution and so are the functions

   for every . So there are no unique solutions. Moreover is not a maximal solution.