This DE is solvable exactly...
where
Hi good people, I'm doing some old exam questions and here's one I don't quite get.
We have the ordinary diff. eq,
where and .
If we want to solve this by using Euler's implicit formula, we need solve an equation by say fixed-point iteration in each step. How small must the step size be in order for the fixed-point iteration to converge on each step.
First of all, the problem does not say what kind of implicit formula I need to use. I get to the formula in the following manner,
,
and so,
.
To get an implicit formula from this I could simply approximate the integral by,
,
or I could approximate it by the Trapezoidal rule.. Let's assume I do the former such that,
Now I need to use fixed-point iteration to solve this equation. If I let my first guess at be which I believe is a reasonable guess, then I could write this thing in the familiar way;
.
The sequence defined by the expression above converges if for all in the interval. But and so but that doesn't make much sense does it? I don't think the step size must be zero Hope someone can help me out here.
Thanks.