- Getting started
- Arithmetic
- Algebra
- User-defined Functions and Expressions
- Plots
- Differentiation and Integration
- Differential Equations
- Maple Worksheets
- Printing your Maple Worksheet
- Saving your Maple Worksheet
- Opening a Previously Saved Worksheet
- Ending your Maple Session
- Obtaining Copies of the Labs in Worksheet Form
- Useful Commands and Techniques
- Getting Help From Other Students

Since you will not be given a Maple manual, you should learn to use
Maple by using the built in **Help** facility. To get help on a particular
topic, such as the `plot` command, type, after the prompt >, the
command `?plot;`. A window will open describing the basic structure
of the command and giving examples of its use. To close this window when
you have finished with it, click the left mouse button on the rectangle
in the upper left corner of the window and then drag the mouse pointer
down, releasing the button when **Close** is highlighted. Another way
to invoke **Help** is to place the mouse pointer on the word **Help**
at the far right of the Menu Bar, click the left mouse button, and drag
the mouse pointer down, releasing the button when the desired item (such
as **Topic Search**) is highlighted. We shall refer to this process
as *choosing* **Topic Search** from the **Help** menu. Fill
in the **Topic** box, click the left mouse button on the item that you
want, and then click on **OK**.

To continue, you will need to know some basic commands and syntax of Maple.

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(2 x +y

`(2*x +y^2)/(2*x + exp(x)) + 1;`

Note that the exponential function is built into Maple and is referred
to as `exp`. Similarly, Maple knows the functions `log`,
`sin`,
`cos`, `tan`, and many more standard functions.

When operating on integers, Maple does exact arithmetic, rather than
using decimal approximations. To get a decimal approximation, use the Maple
command
`evalf`. The Maple command `evalf(4/7,20);` produces
a 20 digit approximation to 4/7. Typing only `evalf(4/7);` will
produce a 10 digit approximation -- as will typing `4.0/7.0;`.

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The command for both definite and indefinite integration is `int`.
If Maple cannot evaluate a definite integral exactly, numerical integration
may be used. Type `?int` and `?int[numerical]` for details
on integration in Maple.

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Consider the differential equation d^{2}y/dx^{2}-y=x^{3}.
To describe this equation in Maple, we need to communicate that the unknown
solution y is a function of x, and to describe the relevant derivatives
of y with respect to x. To communicate to Maple that y is the dependent
variable and x is the independent variable, we write y(x) instead of y
when referring to this variable. Since the first derivative is `diff(y(x),x)}`,
the second `diff(y(x),x$2)`, etc., the differential equation d^{2}y/dx^{2}-y=x^{3}
should be expressed to Maple as `diff(y(x),x$2) -y(x)=x^3` . Notice
that Maple echoes it back in symbolic form. To use this equation later
we give it a name by typing `de:=diff(y(x),x$2) -y(x)=x^3;` so that
the symbol `de` will refer to this equation. We can use the commands
`rhs`
and `lhs` to refer to the right or left side of the differential
equation. To place other conditions on y(x), we group additional equations
together with `de` inside braces. For example, the initial value
problem d^{2}y/dx^{2}-y=x^{3}, y(0)=1, y'(0)=2
can be described to Maple via `ivp:={de,y(0)=1,D(y)(0)=2};` . Note
the use of D as another notation for differentiation. Any conditions on
higher derivatives at a point are described via the D notation as well,
for example `D(D(y))(0)=3`.

Once a differential equation has been described to Maple, Maple can
attempt to find its general solution, particular solutions, or to plot
solutions. We briefly describe the relevant functions here. Find more about
a function by typing `?functionname`. For example `?dsolve`
will tell you all about the Maple function `dsolve`.

**Brief summary of relevant Maple commands for Differential Equations**

`dfieldplot` produces a plot of the direction field of a single
first order differential equation or a system of two first order autonomous
differential equations.

`dsolve(de,y(x));` or `dsolve(ivp,y(x));` In these basic
forms, `dsolve` will attempt to solve the differential equation
or the initial value problem for y(x) described in `ivp`, returning
a formula for y(x) if possible. In the first form, arbitrary constants
`_C1`,
etc. will be used to express a general solution.

`soln:= dsolve(ivp,y(x),numeric);` In this form `dsolve`
finds an approximate solution to the initial value problem described in
`ivp`.
The output `soln` is a Maple procedure. To get the value of this
approximate solution at a particular point, e.g., x=2.5, type `soln(2.5);`.
Because the solution is returned as a Maple procedure which produces output
of the form `[x= 2.5, y(x) = 6.437]`, the
`plot` command
cannot be used directly to graph the approximate solution. Instead, use
the `odeplot` command. Sometimes it is useful to convert the approximate
solution from the form of a Maple procedure into a standard function that
just returns the y-value. This can be done by typing `yapprox:= u->
subs(soln(u), y(x));`. A graph of the approximate solution over the
interval 0 < x < 5 can then be obtained by typing
`plot(yapprox,
x=0..5);`.

`odeplot` is used to plot numerical solutions of differential
equation obtained from using the `numeric` option of `dsolve`.

`DEplot` may be used to plot numerical solutions of differential
equations with a set of initial conditions. In the case of a single first
order equation or a system of two first order autonomous equations, it
can also be used to plot the direction field of the equation, with or without
solution curves. In fact, the default option is to also produce the direction
field; to suppress the direction field, one must add the option `arrows
= NONE`. For computing the numerical solution at a particular point,
the
`numeric` option of `dsolve` should be used. The default
numerical method used by the `DEplot` command is the classical fourth
order Runge-Kutta method with a fixed step size. For some problems, the
default step size may be too large to produce an accurate solution. This
may be corrected by using the `stepsize` option. In general, the
default method used by `DEplot` is a poor choice of method, so it
is better to use the command with the option `method=rkf45`, which
is the default method for the `numeric` option of the `dsolve`
command.

**Examples of use of Maple commands for Differential Equations**

`with(plots): with(DEtools):`
`de1:= diff(y(x),x) = - y(x) + 1/(1 + exp(x));`
`s1:=dsolve(de1,y(x));`
`t1:=rhs(dsolve({de1,y(0)=-2},y(x)));`
`plot(t1,x=-1..5);`
`dfieldplot(de1,y(x), x=-1..5,y= -6..4);`
`initval:={[y(0)=-2],[y(0)=1]};`
`DEplot(de1, y(x), x=-1..5, initval,y=-6..4);`
`t2:=dsolve({de1,y(0)=-2},y(x),numeric);`
`odeplot(t2,[x,y(x)],-1..5);`

**Linear Algebra:** To use Maple's linear algebra commands, you first
enter `with(linalg):` To see a list of commands in this package,
type
`with(linalg);`, i.e., use a semicolon instead of a colon.
In this course, we will only use a few of Maple's linear algebra commands:
`matadd`
to add two matrices, `multiply` to multiply a matrix times a vector,
`linsolve` to solve the linear system of equations A x = b, `det`
to find the determinant of a matrix, and `eigenvects` to find the
eigenvalues and eigenvectors of a matrix.

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The window which appears when you start Maple is called a **Maple worksheet.**
To complete each computer assignment in Mathematics 251 you are asked to
turn in an edited printout of your work; such a printout can be obtained
by editing, and then printing, your worksheet. The printout should include
only the numerical, symbolic, and graphical output of Maple which is appropriate
for the solution of the problems assigned, plus text material in which
labels are provided for graphical output and explanations added. Maple
includes various editing capabilities which should enable you to produce
neat and coherent output, and which we now describe.

To remove an unwanted portion of your Maple worksheet (e.g., a region
containing commands that you typed incorrectly or that were not directly
relevant to the solution of the exercises), select the region to be deleted
by clicking the left mouse button at the beginning, then dragging the mouse
across to the end of the portion of the worksheet you wish to delete. The
region should now be highlighted. Then choose **Cut** from the **Edit**
menu. To copy a region to a new location, select the region as above, but
now choose **Copy** from the **Edit** menu. Then click the left mouse
button in the position in which you wish to insert your selected region
and choose **Paste** from the **Edit** menu.

To insert text, such as a label for a plot, into your worksheet, click
the mouse at the beginning or end of the plot and choose **Text Input**
from the **Insert** menu. Now type your label. To continue using your
worksheet, move the mouse pointer down to the next prompt > and click the
left mouse button. If there is no prompt, insert one by clicking on the
prompt symbol > in the Tool Bar Menu.

Sometimes it is useful to be able to place a comment after a Maple command, rather than insert text elsewhere in the worksheet. To do this, enter the sharp symbol #. Everything typed on a line following this symbol will be considered by Maple to be a comment, and therefore not executed.

To make your worksheet less cluttered, it is a good idea to have Maple
suppress the output of various commands, e.g., the command `with(plots)`
or a command given to assign a name to a plot. To do this, end the command
with a colon (:), instead of a semicolon (;).

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The home page of the course contains a section that contains pointers to these Maple worksheets. Follow the instructions there to obtain your personal copy of the worksheet before starting Maple.

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Any Maple command previously entered in your worksheet can be re-executed
without retyping it in a new location. Simply move the mouse to the position
of the command you wish to execute and hit the **Return** key.

It is often useful to be able to refer later to the result of a computation
-- the output of some command -- in a simple way. To make this possible,
simply assign the output of the command to a variable. For example, if
you enter
`a:= evalf(2*Pi);` then you can later square the result
of `evalf(2*Pi);` by typing `a*a;`.

You can assign a name to a plot just as described above for assigning
a name to an expression. Several previously named plots can then be displayed
on the same graph by using the command `display`. Type `?plots[display]`
for details.

If you assign the name `a` as above and then continue your Maple
session, you may want to reassign the name `a` to another expression.
To do so, first unassign `a` by typing `a:='a';`. To clear
all the assigned variables in a Maple session, type `restart;`.
One common problem is to try to use such a variable without unassigning
it, forgetting that is has already been assigned a value. A simple way
to avoid this is to issue the `restart `command before beginning
a new problem. This will not change anything on your screen.

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Last modified January 5, 2000 by *Richard T. Bumby, bumby@math.rutgers.edu,
*based on the version of January 6, 1999 by
the original author:
*Richard S. Falk, falk@math.rutgers.edu*

Comments on this page should be sent to: bumby@math.rutgers.edu