of
Since the differential equation is non-linear, the existence and
uniqueness theorem guarantees only that there is a solution in some
interval about t = 0. Suppose that we try to compute a solution of the
initial value problem on the interval 
using different
numerical procedures. If we use the Euler method with 
, and
, we find the following approximate values at t = 1, 
and 
, respectively. The large differences among the computed values
are convincing evidence that we should use a more accurate
numerical procedure---the Runge-Kutta method, for example.
Using the Runge-Kutta method with 
we find the approximate value
at t = 1, which is quite different from those obtained using the
Euler method. Repeating the calculations using step sizes of 
and
, we obtain the interesting information listed in Table 2.
Table 2: Calculation of the Solution of the Initial Value Problem 
, 
, using the Runge-Kutta Method
While the values at 
are reasonable and we might well believe that
the solution has a value of about 
at 
, it is clear that
something strange is happening between 
and 
. To help
determine what is happening, let's turn to some analytical approximations to
the solution of the initial value problem. Note that on 
,
This suggests that the solution 
of
and the solution 
of
are upper and lower bounds, respectively, for the solution 
of
the original problem, since all of these solutions pass through the same
initial point. Indeed, it can be shown that 
as long as these functions exist.
This is important because we can solve Eqs. (4) and (5)
analytically for 
and 
by separation of variables.
We find that
thus 
as 
, and
as 
.
These calcuations show that the solution of the original initial value
problem must become unbounded somewhere between
and t = 1. We thus see that the problem (2) has
no solution on the entire interval 
.
Our numerical calculations, however, suggest that we can go beyond
, and probably beyond 
. Assuming that the solution
of the initial value problem exists at 
and has the value 
,
we can obtain a more accurate appraisal of what happens for larger t
by considering the initial value problems (4) and (5)
with 
replaced by 
. Then we obtain
where only four decimal places have been kept. Thus
as 
and 
as 
.
We conclude that the solution of the initial value problem (2)
becomes unbounded near 
. We cannot be more precise because
the initial condition 
is only approximate. This example
illustrates the sort of information that can be obtained by a
judicious combination of analytical and numerical work.
