### 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?