**Log in Campus network**

To use Matlab program, you first need to log in W&M Campus network.
The location of Labs can be found at __http://www.wm.edu/IT/labs/map.html.__
Here is the procedure to open Matlab:

- Log in the computer network using your W&M ID and password(usually p and last 6 digits of your social security number). For the domain, choose "Campus".
- Once Windows 2000 has started, click "Start"->"Programs"->"Courseware"->"Math"->"Matlab 5.3"->MatlabR12"

It will take a few seconds to 30 seconds for Matlab to initialize.

**Basic Commands of Matlab**

Matlab is a commecial mathematics software which can do a lot of computation
and visualization of mathematics. Most of time we only need to use John
Polking's programs: dfield and pplane. But in the lab we will also use
some Matlab functions to do Euler's method, data fitting and others.

To start dfield, simply type

>> dfield6

and the windows for dfield will pop up. And the command for pplane
is

>> pplane6

Next we show some other Matlab functions by examples.

**Plotting function graphs**
**Example**. Sketch the graphs of y=x^2-3x+5 and z=x^3+6x^2-6
over the interval [-2,3] on the same figure. Use a solid line type for
the first graph and a dashed line type for the second graph.

>> x=-2:0.05:3;

>> plot(x,x.^2-3*x+5,x,x.^3+6*x.^2-6,'--')

**Euler's method**
**Example**. Use Euler's method to plot the solution of the initial
value problem*
y'=y+t, y(0)=1*
*on the interval [0,3].*

Before using the routine for Euler's method, you have to first write
a function M-file for f(t,y)=y+t.

In Matlab main window, click "File"->"New"->"M-file", then a new window
to write a M-file in will show up. In that window, type

>>function yprime=yplust(t,y)

>>yprime=y+t;

then save the file in a directory of your choice. For example, you
can save it to C:\my files\M-files\

and the filename is yplust.m. Then you have to add this directory to
the paths which Matlab will search for excutables. To add the path, click
"File"->"Set Path", and a new window pop up. In the new window, click "Add
folder", and choose C:\my files\M-files\, then click "Save". Now try type
this in the main Matlab window:

>>yplust(1,2)

and you will get

ans =

3

Now we use Euler's method routine. Type

>> [t,y]=eul('yplust',[0,3],1,0.1);

>> plot(t,y)

A graph of the approximate solution will show up. Here [0,3] is the
interval to perform Euler's method, 1 is the initial value y(0)=1, and
0.1 is the stepsize. To just plot the data points, you type

>>plot(t,y,'o')

to disply all data points, simply type

>>y

**Data Fitting**
**Example. **[PBA] page 135 problem 8.

First we need to enter data points. In Matlab, this can be done using
vectors. Here let's try for the first five data points in the problem.
Type

>> t=[0,1,2,3,4]

>> p=[10,14,19,24,28]

>> f = polyfit(t,p,1)

then f would be the linear function which fit the data set best. Type

>> f

you would get

f =

4.6000 9.8000

That means p=4.6*t+9.8. To plot the data points and the regression
line, type

>> t1=0:0.1:4;

>> p1=polyval(f,t1);

>> plot(t,p,'o',t1,p1)

**Exercises**

1. In dfield, draw the direction field for P'=P(1-P)-aP cos(t) with
a=0.3