Math 321.001: LAB 1

Euler's Method Lab

This lab is due Friday,September 30, 2016 in class. Late labs will not be graded. You may use any technology that you have available: excel, Wolfram Alpha, Matlab, etc. I recommend excel because you can easily use it to produce tables and graphs that you can then paste into a word docuement. 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.

**Group Work Rules:** This lab should be completed by a group of
3 to 5 students. Turn in one copy of the completed assignment
with the names of all group members. After the graded assignment
is returned, copies should be made so that each group member has a copy
to keep.

** Introduction. **In this lab, you will need to use two
numbers, **A** and **B**.
These numbers are derived as follows. Choose the student id
number for one member of the group. The number **A** is the last **nonzero**
number in that student ID, while the number **B** is the next to
the last **nonzero** number in the ID. For example, if the student
ID
is 1454809, then **A** = 9 and **B** = 8.

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 as described 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 grading will
stop 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) = 1+B/10
**

where the constant **B** is determined from the 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, grading stops 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, give a qualitative description of how your
results from the first and second parts of the lab compare. Do
they show a similar improvement of Euler's method with decreasing step
size?