MA 231: LAB 1

Another Numerical Method Lab

This lab is due Tuesday, September 23, in class. Late labs will not be graded. You may use any technology that you have available: a spreadsheet, Mathematica, Matlab, programmable calculator, etc. You will be graded on exactly what is asked for in the instructions below. You need not turn in any additional data, graphs, paragraphs, etc. You should submit only what is called for, and in the order the questions are asked. Remember that you will be graded on your use of English, including spelling, punctuation, logic, as well as the mathematics.

**IMPORTANT:** The work you submit should be your own and nobody else's.
Any exceptions to this will be dealt with harshly.

Your goal in this lab is to develop a numerical algorithm for approximating solutions to differential equations that works a little better than Euler's method. After you develop the required formulas, you will then compare the results to those obtained by Euler's method for a given differential equation and for several different step sizes.

Here is a qualitative description of the new method. READ THIS CAREFULLY. Your job is to translate this description into mathematical formulas. Suppose you start with the initial value problem

Your goal is to obtain an iterative scheme just as in Euler's method
that produces a sequence of values **(t _{n},
y_{n})**
that approximates the
solution to the given initial value problem.

As with Euler's method, the new method begins with
**t _{0} = 0** and

To obtain **y _{n+1}**
we modify Euler's method as follows. Be careful: you must read this
very carefully to succeed.
We will draw a straight line
through the point

That is, the second straight line has slope given by the
slope field, not at **(t _{n},y_{n})**,
but rather at the point on the
first line directly over

1. This question should be answered **BEFORE** turning to
technology.
First give the formula you use to obtain **y _{n+1}**
for the numerical
solution of the initial value problem

**WARNING:**
Make sure that your answer here conforms
to the instructions above. Otherwise it will be impossible to go on,
since you have the wrong formula.

2. Now use the formula you derived in question 1
to approximate the value of **y(1)**
the initial value problem

3. Now repeat the previous question, this time using Euler's method rather than the new method.

4. What is
the actual value for **y(1)**? That is, find the real solution to
the differential equation and then compute **y(1)**

5. Which numerical method yields the better approximation?

6. Now repeat questions 2 and 3, this time with **Delta t =
0.01**. This time you should calculate **y _{0}**
through

7. Now compare the errors made in both numerical solutions. In an essay, discuss the question: How does changing the step size in both methods affect the error? More precisely, if you change the step size by a certain factor, how does the step size change in each method? You will need to perform several more experiments similar to those in questions 2-6 above. Use the same differential equation but different step sizes.