MA 226: LAB 1

Euler's Method Lab

This lab is due Tuesday, September 26, 2017 in class. Late labs will not be graded. You may use any technology that you have available: a spreadsheet, Mathematica, Matlab, 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.

** Introduction. **In this lab,
you will need to use two numbers, **A** and **B**.
These numbers are derived from your student ID (**not** your Social
Security number)
as follows. The number **A** is the last **nonzero**
number in your student ID, while the number **B** is the second last
**nonzero** number in your ID. For example, if your student ID
is 123-45-6789, then **A** = 9 and **B** = 8. But if your ID
is 100-20-3000, then **A** = 3 and **B** = 2.

We have seen in class how to use Euler's Method to approximate the solutions
of differential equations. We have also seen that Euler's method usually
increases in accuracy if more steps are used (equivalently, if ** Delta t**
is chosen smaller. In this lab you will investigate how the accuracy
of Euler's method changes as the step size becomes smaller.

Answer each of the following questions in order

**1.** Consider the initial value problem

** **

y(0) = A

where the constant **A** is determined from your student ID as above.
Find the exact solution **y(t)**
to this initial value problem and
determine the value **y(1)**. Be sure to check that your
answer here is correct and show this computation explicitly. If your answer
here is wrong, the rest of this lab makes no sense and we will stop
grading at this point.

**2.** Use Euler's method with a step size of **Delta t = 0.1 **
to approximate **y(1)**. That is, using Euler's Method, compute
in succession

where

**3.** Repeat question 2 with a step size of **Delta t = 0.05**,
i.e., with twice as many steps.

**4.** Repeat question 2 with a step size of **Delta t = 0.01**,
i.e., with ten times as many steps as in question 2. You need not
present all of the data here; just give the approximation to **y(1)**
that you find using this step size and the error.

**5.** In a brief essay (no more than one page),
discuss the improvement of the accuracy
of Euler's Method as you make the step size smaller by a factor of 1/2 and
1/10. How does this affect your approximation of **y(1)**? By how much
does your approximation improve percentage-wise?

**6.** Now consider a second initial value problem

** **

y(0) = B/10

where the constant **B** is determined from your student ID again
as above. Remember to use **B/10**, not just **B**.
Now repeat questions 2-5 for this initial value problem. Remember to
check first that your exact solution
of this initial value problem is absolutely correct (using this value).
Otherwise, we
stop grading at this step.

So that everyone starts at the same place, consider the value

Your results to the second part will not be as "clean" as in the first part. In your essay, discuss how close these two different results are.