A differential equation
may be approximated as a difference equation.
If
, then
Iterating the approximation
, we can
numerically approximate solutions to initial value problems
and
.
That is, given that satisfies the above initial value
problem, to approximate
, fix a positive integer
,
set
, and define
(for
).
We know that . Approximating, we have
Repeating this process, we find that
,
...,
.
Approximate the value of when
and
using
.
Note that a symbolically solve one must
find an antiderivative to
.
Here
.
We compute
Approximate when
and
using
subdivisions.
This time, our symbolic methods fail twice! To use
the method of separation of variables, we would need to
find an antiderivative of . Even if we were to
succeed with this step, we would have to invert the function
.
In this case, we compute mechanically.
,
, and we wish to find
.