(due: May 6 in class)

**Problem 1 **(20%):

Starting from equation 5.30 in the textbook, derive the expression for the fourth order Adams-Bashforth technique
(equation 5.24).

Problem 2 (30%):

Consider the following initial value problem:

dy/dt = -20*y + 20*t^2 + 2*t

0 <= t <= 1

y(0) = 1/3

Assume you want to solve this problem with the Trapezoidal Method. Show that applying Newton's Method leads to
the equation

w_{j+1} = w_{j} *(1-10h)/(1+10h) + h/(1+10h) * [10*(t_{j})^{2} + 10*(t_{j+1})^{2}
+ t_{j} + t_{j+1}]

**Problem 3 **(50%):

Consider the initial value problem in problem 2.

The exact solution to this problem is y(t) = t^2 + 1/3*exp(-20*t)

Solve this problem numerically using

a) the Trapezoidal Method (cf. problem 2)

b) Euler's Method

c) the standard Runge-Kutta Method of order 4

Use the timesteps h=0.2, h=0.12, h=0.1, and h=0.02 for all methods. Compare the results; in particular, compare
which methods become unstable, and make a statement about the regions of absolute stabilty based on the values
of h that were chosen.

**Hint**: You should "recycle" your old scripts for the Euler Method and RK4.

**You should turn in the following: **

- Plots that show the different solutions (and the exact result) as a function of t for all h;

- Interpretation/Discussion/Description of your results.