The DFIELD Tool
Now I want to introduce you to a nice online tool which might
help you construct some of
these slope fields. (if you get tired of doing this by hand and
your calculator can't plot these for you) It is called DFIELD
(named for the term direction field which is another term for
slope field). It is being developed by
John C. Polking
and some of his colleagues at Rice University. The software is really
just a set of m-files that
MatLab executes. MatLab is a Matrix Laboratory software package that is
very popular with many mathematicians and engineers. As a matter of
fact, the image that you see on the right of this page was generated
using DFIELD.
Before you actually visit the site, you should
remember that there are a few syntax rules that you need to understand and follow.
After all, this is a computing tool, much like a calculator,
and it requires a certain syntax in order to function properly.
You may want to keep this window open while you are using the
tool so that you can read through the following notes as you plug
along for the first time. Just minimize it and pull it up whenever
you need it. The link to DFIELD is at the bottom of the
page, but read (at least skim it, please) through the following
before you click.
- There will be several windows that pop up as you get started.
As the software is cranking up, you will have to click OK
on the copyright box in order to
get started. Managing all of these windows will require some
ingenuity on your part.
- Two Basic Windows:
- The two basic windows that you will work with are the
Equation Window and the
Direction Field Window.You
may have to adjust the size of the Equation Window in order to
see the contents of all the boxes. The Equation Window is where
you enter the differential equation. Notice that there is already
an equation entered for you. There is a box which tells DFIELD
which variable is the independent variable.
There are also some parameters in the equation-- a and b. The
values of those parameters can be adjusted in the boxes under the
heading Parameter Expressions. One
can also adjust the maximum and minimum values of the graph of
the slope field by using The Display
Window boxes.
- Plotting a slope field:
- Click on the button in the right hand
corner of the Equation Window. It is labelled as
Graph Direction Field. You
can view the direction field (or slope field) in the
Direction Field window.
- Sketching Solution Trajectories:
- Using your mouse to click on a point in the
direction field, you can generate a sketch of the solution
to your differential equation which passes through that
particular point in the plane.
- When you click on a particular point in the
ty-plane, you are, in essence, specifying an initial
condition for your differential equation. DFIELD
then sketches that particular solution for you.
- Basic Syntax and Operations:
- addition, +
- subtraction, -
- multiplication, *, (don't forget
this one: 3x is written as 3*x)
- powers, ^, (3^2 means three
squared)
- Functions:
- trigonometric, sin(3*x),
(the parentheses must be there, `sinx'
will only give you syntax errors and frustrate you) and it
works the same for cosine (cos(3*x)) and tangent (tan(3*x))
functions
- the exponential function e, exp(x) or exp(t^2),
(again the
parenthese are the big deal)
- Quitting: You may quit the application by clicking the
Quit option under the File menu
in the Direction Field Window.
Click Here
in order to use DFIELD. Note: There is
some introductory stuff at the top of the page. In order to crank up
the tool, you must scroll down the screen and click on the grey button
labelled DFIELD 2005.10 Also BE AWARE that this link appears in a new
browser window. Organization of your desktop may be required!