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.