### Homework assigned 12/8, due 12/18 at 5 pm

### Where you are asked to present graphs, look at the class diary
from 12/10/98 and save the figures as *.ps files. If you have
problems figuring out how to print *.ps files, make sure these files
are clearly titled with the problem and what you want me to read and
understand. Either give the *.ps files to me on a disk or put them on
your web-page in a way that I can download them.

### I'm moving my Thursday office hours from 10:30-11:30 to
4-5

### Section 9.2

Problems 1b,2a,2b

For #1b, only compute up to x=1.

For #2a, present one graph that contains of y(x,h=.5),
y(x,h=.25), and y(x,h=.125). Present another graph that contains
the errors err(x,h=.5), err(x,h=.25), and err(x,h=.125).

For #2b, present the same types of graphs that you presented for #2a.

### Section 9.3

Problems 1,4,7

For #1, you refer to table #1 in section 9.2, which is for the
problem Y' = -Y, Y(0)=1. On the other hand, in section 9.3, you have
equation (9.41) which tells you that the error should be bounded by
h/2(exp(b)-1). In fact, you can compute that upper bound for various
values of h and b. Do this and make sure that the errors in table
1 really are bounded by h/2(exp(b)-1).

For #4, you're asked to do the same thing: compute the upper bound
of equation (9.38) and make sure that it bounds the errors that you
find computationally. For this problem, _also_ find the ratios of
the errors and make sure that they're behaving as you'd expect them
to behave.

For #7, find the asymptotic bound at points x=2,3,4,5,6 and compare
the bound to the numerical error you find in computing the solution
using Euler's method.

### Section 9.4

Problems 2b,6,7,13

For #2b, only compute up to x=1. Present the errors at x=.2, .4, .6,
.8, and 1

### Nonlinearities make life interesting

Consider the coupled system:

dx/dt = y - (x^2+y^2-1) x

dy/dt = -x - (x^2 + y^2- 1) y

1) Find the fixed points (the steady-state solutions). Find analytically
using pen and paper or maple.

2) Study the system numerically using second-order Runge-Kutta to do
the time-stepping. First, give graphs demonstrating the behavior near
the fixed point(s) (one graph per fixed point). Second, give a
graph demonstrating more global behavior.

Consider the coupled system:

dx/dt = y + (x^2+y^2-1) x

dy/dt = -x + (x^2 + y^2- 1) y

Repeat the above three problems.

Consider the coupled system:

dx/dt = y + a (x^2+y^2-1) x

dy/dt = -x + a (x^2 + y^2- 1) y

where a is some fixed real number.

1) Find the fixed points (the steady-state solutions).

2) Find the behavior of solutions that start near the fixed points.

3) Can you find a geometrical explanation for the behavior of the solutions
for different values of a?