First of all we need to say a little about what differential equations and initial-value problems are, and the conditions under which they have unique solutions.
Then we need to realize that, because we're solving these problems numerically, we're not going to be solving the initial-value problem, but rather one close to that given.
In this section we get a brief view of conditions under which a ``perturbed'' (slightly disturbed) problem will give reasonable information about the real problem.
-
differential equation: an equation linking a function y and its
derivatives and independent variables.
-
initial-value problem: with time as the independent variable, a
differential equation as well as enough initial conditions to uniquely
determine the solution.
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well-posed initial-value problem: the initial-value problem
is well-posed if
- A unique solution y(t) exists, and
-
For any
there exists a positive constant 
such that
whenever 
and 
is continuous with
on [a,b], a unique solution z(t) to
exists with
The problem (2) is called a perturbed problem associated with the problem (1).
-
Lipschitz condition: A function f(t,y) satisfies a Lipschitz
condition in y on a set
if a constant L>0 exists with
-
A set is said to be convex if whenever
and 
belong to D and 
is in [0,1], the point
also belongs to D.
Theorem 5.4: Suppose that 
and that f(t,y) is continuous on D. If f satisfies a Lipschitz condition
on D in the variable y, then the initial value problem (1) has a
unique solution y(t) for 
.
Theorem 5.3: If f(t,y) is defined on a convex set . If a constant L>0 exists with
then f satisfies a Lipschitz condition in y on D with Lipschitz constant L.
Theorem 5.6: Suppose that
and that f(t,y) is continuous on D. If f satisfies a Lipschitz condition
in y on D, then the initial value problem (1) is well-posed.
So the up-shot is that our initial-value problem satisfies a Lipschitz condition, we're in good shape: we're going to be able to get an approximation numerically.