Instructions for Using Winplot

Winplot is a graphing utility written and maintained by Richard Parris, a teacher at Phillips Exeter Academy in Exeter, New Hampshire. The program is "totally free" and the newest version can be downloaded from Parris's web site at http://math.exeter.edu/rparris/winplot.html . All of the Winplot modules have fairly thorough Help menus which give detailed information about the workings of the program. An online power point tutorial has been written by John Ganci.
 


Upon launching Winplot you will see the following introductory screen.

By clicking on Window you will be presented with the following two plotting options:

2-dim     Allows you to generate plots in the x, y plane.
3-dim     Allows you to generate plots of surfaces or space curves in x, y, z space.
Guess     Gives you a graph; you then guess the equation. If you choose Select in the guess my equation box shown below
               you can select the type of graphs that will be displayed.

If Open last file is checked, then for any option chosen the last file saved will automatically appear in the chosen window. If Use defaults is checked a standard viewing window for each type of plot will be initialized. Mapping provides a way to visualize the graph of a function that maps two-dimensional space to two-dimensional space. This is especially pertinent for calculus students as they study integration in two dimensions.

Graphs in 2D

Changing the viewing window

From the introductory screen choose Window 2-dim . You will then see a window containing an x, y grid. The appearance of this viewing window can be altered using the View pull down menu.

To change the plot window use the command View. This brings up the following menu.

If  the upper right "radio button" or check circle entitled "set corners" is selected you can set the window's lower left (left, down) and  the upper right (right, up) coordinates by just entering the appropriate values.Click on the upper right "radio button" or check circle entitled "set corners". This allows you to set the window's lower left (left, down) and the upper right (right, up) coordinates by just entering the appropriate values. In reference back to the first View menu, if the scales on the two axes are not equal (so that circles don't look circular), you can use the command sequence Zoom/Square to make them so. Click Fit window to make all active examples fit within the window. Because this might not be possible, results are sometimes unpredictable. Restore returns to the default window settings. Axes selects the axes menu which allows you to control how x and y axes are displayed.

The Grid dialogue box allows you to display a Cartesian or polar grid by clicking the appropriate radio button. If "both" is selected both x and y axes are displayed. Checking either "x" or "y" displays only the axis selected. If "polar" is selected you can specify the number of sectors by clicking on the square to the left of "polar sectors" and entering the number to the right.

Check "labels" to put the letters 'x' and 'y' into the diagram. If you check "arrows", you get small arrows at the ends of the axes. Check "ticks" to put equally spaced marks on the axes. The distance between successive ticks is the value of "interval". Click on "scale" and "mark scale on axes" to put the corresponding numerical values next to the ticks. Click on "scale" and "mark scale on border" to place the numbers at the edge of the window. The number of decimal places displayed is entered as "places". If there are too many values showing, enter a higher number in the corresponding "freq" box. Check "pi" if you want the ticks labeled using multiples of pi. For this last option to be effective, the corresponding tick interval should be a rational multiple of pi with a small denominator and the font in the "Scale on axes" option of the Misc/Fonts menu should be set to "Pi symbol".  If "rectangular" is checked, grid lines appear in the quadrants checked. If "dotted" is checked, you will see an array of small crosses where the grid lines would meet. The polar grid can be on at the same time as the rectangular grid. To effect the changes of this menu press the "apply" button.

You can insert text into the graphing window by use of the mouse buttons. First use the "Btns" pull down menu to make sure that the Text option is checked. Second place the mouse pointer at the spot where you want to enter text, click the right mouse button, and type in the desired text. If you wish, you can change both the color and font of the text.

To insert a circular dot at a point go to the Equa pull down menu and select Point. In the dialogue box give the point's coordinates. The value of "dot size" determines the diameter of the dot and the color can be adjusted via the color button.

Plotting the graphs of equations

To see a graph in Winplot you must either Open an old Winplot file (files with the extension wp2), or else enter a new equation by clicking on the Equa menu. There you will be presented with the following four options for generating curves.

1) Explicit        Graphs y as an explicit function of x .

2) Parametric   Graphs parametric curves of the  form x = f (t ),  y = g(t ) for f and g explicit functions of some parameter t .
                        By letting g(t ) = t this in effect allows you graph x as an explicit function of y, i.e., x = f (y) .

3) Implicit         Graphs implicit relations between x and y defined by a formula f (x, y) = constant.

4) Polar            Graphs polar curves where the polar coordinate r is an explicit function, f(t), of the polar angle theta (here
                        designated as t).

Clicking Explicit from the Equa menu results in the following dialogue box.

By typing over the "f(x) =" box you can enter an expression that defines y as a function of x. Click the "color" button to obtain the color palette and click the small box that shows the color you want for the graph. You can type an integer into the "plotting density" box to alter the density of plotted points. This is rarely necessary. If you want to specify the domain of the function enter the interval in the boxes "low x" and "high x" and then click on the box "lock interval". This overrides the default which is to use the interval defined by the plot window as the domain. More than one graph can be displayed in the same graph window. Just return to the Equa menu and repeat the procedure outlined above. In fact, you can mix graphs of y as a function of x with polar curves, parametric curves, and implicit curves by simply using the different options for entering equations.

By choosing Inventory from the Equa menu you access the inventory dialogue box. This allows you to inspect and edit any of the equations already entered and perform other modifications and constructions. To select an equation, click on it with the mouse. Only one example can be selected at a time.

The Inventory options include the following:

edit - This button opens the dialogue box that was used to create the curve and allows you to make changes.

delete - This button causes the curve to disappear from the inventory and from the screen. There is no "undo".  All equations that are dependent on the equation are also deleted.

dupl - This button copies the selected equation and opens the editing dialogue. This allows you to create a similar example without changing the original. If you want to keep the original as well as the copy be sure to click no to the prompt "delete original?".

clip - This button copies the text of the defining equation to the clipboard where it can be pasted into other applications.

derive - This button calculates the derivative of the selected curve. This option only applies to curves defined by the Explicit, Parametric or Polar options. The result is graphed and added to the inventory. A derivative can therefore also be selected, but only its attributes (color, thickness, etc.) can be edited.

name - This button gives a  selected equation a name which appears in the inventory.

hide graph - This button hides the graph of a selected equation without removing it from the inventory. A second click restores the graph.

show equa - This button displays the text of a highlighted equation in the upper left corner of the graph window. A second click removes this display.

web - This button overlays a web diagram which is an iteration used to locate fixed points of a curve of the type y = f(x). You can specify the starting x-value, the number of steps, and the color of the overlay. To generate the web diagram select "define". To delete the web diagram select "undefine".

family - This powerful feature allows you to display a family of curves generated from a selected equation. The equation must be defined with one or more extra parameters. For example, the equation y = sin(ax)+b defines a sine wave that depends on the two parameters a and b. Either one of these can be used to create a family of curves. To vary the frequency type "a" into the "parameter" box, enter a range of values by filling in the "low" and "high" boxes, and indicate the number of curves to be drawn by filling in the "steps" box (this is one fewer than the number of curves). Click "define" to generate the family. This also modifies the definition of the equation in the inventory. To remove the family, select the equation and click "undefine".

table - This button opens a text window that displays values of the selected function for curves defined by the Explict, Parametric, or Polar options. You can alter the contents of the table by clicking "Params" on the menu bar. To see tables for a different equation use the command sequence "File/Next example".

Winplot like a graphing calculator has built-in "Zoom" features. To "Zoom in" and thus obtain a close up view of a smaller portion of the x, y plane choose Zoom/In from the View menu. Similarly, to "Zoom out" choose Zoom/Out from the View menu. If you're not happy with the new image choose Last window from the View menu.

When entering formulas in Winplot the following conventions are used:

Multiplication is indicated by *
Division is indicated by /
Addition is indicated by +
Subtraction is indicated by -

Exponentiation is indicated by placing the ^ (caret symbol) between the expression for the base and the expression for the exponent. The caret is the upper case of 6 on the keyboard.

Actually, Winplot does recognize algebraic notation. For example, "two times ex", can be entered either as 2x or 2*x . Similarly, "three times ex cubed", can be entered as 3x^3 or 3*x^3 or 3*x*x*x or even as 3xxx. The only valid grouping symbols are parentheses ( ), but these may be nested as in (6x - x^(sqr(5)+3))/(5x + 2). The rules for the order of operations are the conventional ones used in mathematics and science.

Certain standard functions are "built into" Winplot, but the arguments must always appear in parentheses. For example, "the sine of 3 times x" should be written as sin(3x).

The names of the principal "built in" functions are given below.

abs(x) is the absolute value of x
sgn(x) is the signum or sign function = abs(x)/x
sqr(x) is the square root of x (for non-negative x)
root(n,x) is the principal n-th root of x
fact(n) is n!
exp(x) is the exponential function evaluated at x
ln(x) is the natural logarithm function for positive x
log(x) is the base 10 logarithm function for positive x
sin(x) is the sine of x (All trig functions assume the argument is in radians.)
arcsin(x) is the arc or inverse sine
cos(x) is the cosine of x
arccos(x) is the arc or inverse cosine
tan(x) is the tangent of x
arctan(x) is the arc or inverse tangent of x
sinh(x) is the hyperbolic sine of x
cosh(x) is the hyberbolic cosine of x
tanh(x) is the hyberbolic tangent of x

pi is the constant pi=3.14159265. To multiply pi and x, enter pi*x, not pix.

More information about "built-in" functions can be found in the file Equa/Library.

As an example, suppose you want a graph of the cubic polynomial

From the Equa menu select Explicit and enter the formula as shown below.

Clicking the "ok" button results in the following: (the grid settings used are also displayed)

Suppose now we click Polar from the Equa menu. This results in the following dialogue box for a polar curve. This second curve will be graphed along with the previous polynomial.

The usual theta domain for polar curves will include at least zero to 2pi. Entering 2pi for "high t" is the same as entering
6.28319 . The result of graphing both curves in the graph inventory is shown below.

Suppose we wanted to graph the ellipse centered at (2, -1) with major axes parallel to the x axis, a semi-major axis of 3, and a semi-minor axis of 2. Since y is not a function of x for this figure, a single y = f (x ) graph can not generate the plot. However, we can graph it by using the parametric input mode. Selecting Parametric from the Equa menu and entering the formulas shown below generates the correct result.

The standard equation of this ellipse is  . An alternate way to graph it is to use the implicit input mode. The equation generating the graph must be stated as f(x, y)= constant. The points on the curve are generated by a numerical search. If the "long search" option is checked the program will continue searching for points until you press "Q" (for quit). Failure to type "Q" causes the program "to hang". Implicit from the Equa menu and entering the formula shown below generates the graph shown.

Mathematical calculations and demonstrations

In addition to graphing equations Winplot can perform numerical and graphical procedures on curves. These options are found in the Anim, One, Two and Misc menus.

The Anim menu provides for some of the most dramatic and illuminating demonstrations available in Winplot. In Winplot the letters a ... z always have a numerical value, and letters other than x, y, and z can be used in equations as parameters as was discussed in the Inventory/family option. The value of such a variable can be altered using the scrollbar dialogue boxes listed in this menu. The box labeled by the upper case name of each parameter controls the values of that parameter. When a parameter value is changed, all parameter-dependent graphs, including vector field plots generated under the Equa/Differential option, change accordingly.

The values of a parameter vary from its lowest or left-most value "L" to its largest or right-most value "R". These values can be entered and then set with the buttons "set L" and "set R". Within this domain the value of the parameter can be set manually by moving the scroll bar. This process can be automated by pressing the "autorev" or "autocyc" buttons. The "autorev" option varies the parameter according to the sequence L.R.L:. R. , while the "autocyc" option uses the sequence L.R L.R L.R. In both automatic modes the dialogue box disappears and you must press "Q" to stop the animation and restore normal program control. By default there are 100 intermediate values for the parameter between the L and R values. This number can be changed by clicking on the "1=Scrollbar units" item. In addition, the number of different values of the parameter for which graphs are displayed in "autorev" or "autocyc" mode can be controlled by checking "autoshow" the entering the number of "slides" to be displayed.

The linked window option allows a second graph window with a curve controlled by a parameter to be simultaneously displayed.

As an example, suppose we wish to animate the parameterized function f (x) = sin(a*x) + b . Use the Anim menu and select A and B in succession to set the range for each parameter. In the figures shown below B was set to 0 and A varied from 0.1 to 10.


 

The One pull down menu lists procedures that Winplot can do to one function at a time.
  1. To "trace" along a curve select Slider. A scroll bar appears along with a crosshair cursor that slides along the non-implicitly defined curves in the graphing window. If there are multiple curves in the graph's inventory, use the drop-down list at the top of the dialogue box to select the desired equation. The cursor will move along the curve and the tracer box will display its coordinates as you slide the scroll bar. You can position the cursor by entering a value for the independent variable into the edit box. To save a cursor position in the inventory press the "mark point" button. Use the "degree" drop-down list to set the degree of a Taylor polynomial of a curve of the y = f(x) type calculated at the current cursor position. Press the "Taylor approx" button to display a graph of this approximating polynomial. This Taylor polynomial will then be added to the graph's inventory. If "secant demonstration at" is checked and a fixed point on the curve has been set using the "base point" button, the program will graph a moving secant line to the curve as the scroll bar is moved. If "tangent-line demonstration" is checked, the program will graph a moving tangent line to the curve as the scroll bar is moved.
  1. To find the roots or x intercepts of a graph y = f(x), select the Zeros dialogue box. If there are multiple equations in the graph's inventory use the drop-down list at the top of the dialogue box to select the desired equation. Step through the equation's roots from left to right by clicking on the "next" button. After you have found the roots of a function, you can see a list of these zeros by selecting Data/Inspect from the Misc pull down menu.
  1. To find the extremes (maximums and minimums) of a graph y = f(x), select the Extremes dialogue box. If there are multiple equations in the graph's inventory use the drop-down list at the top of the dialogue box to select the desired equation. Step through the extreme values from left to right by clicking on the "next extreme of " button. After you have found the extremes of a function, you can see a list of these results by selecting Data/Inspect from the Misc pull down menu.
  1. The Measurement menu allows for a variety of numerical approximations to definite integrals.

To approximate integrals of y = f(x) select the Integration dialogue box. If there are multiple equations in the graph's inventory use the drop-down list at the top of the dialogue box to select the desired equation. Enter values into the "lower limit" and "upper limit" boxes and specify a value for the number of "subintervals" (n). Indicate with the check boxes which numerical quadrature methods (parabolic is Simpson's rule ) are to be used. If "overlay" is checked, the program will graphically illustrate the approximation. Press the "definte" button to display the numerical results of the approximations. The number of decimal places shown can be adjusted with the command sequence Misc/Decimal places. The "indefinite" button generates the graph of the antiderivative of f(x) that is zero at the lower limit of integration. This function is then also added to the inventory.

The Area of a Sector option evaluates the area of the sector (centered at the origin) swept out by a selected arc defined by a non-implicit curve. If there are multiple equations in the graph's inventory use the drop-down list at the top of the dialogue box to select the desired equation. The starting and ending values of the independent variable are entered into the "arc start" and "arc stop" boxes respectively. Specify a value for the number of "subintervals" (n) and press the "area" button to obtain the numerical result.

The Length of arc option computes the arc length of a non-implicit curve. If there are multiple equations in the graph's inventory use the drop-down list at the top of the dialogue box to select the desired equation. The starting and ending values of the independent variable are entered into the "lower limit" and "upper limit" boxes respectively. Specify a value for the number of "subintervals" (n) and press the "length" button to obtain the numerical result.

    To compute a volume of a solid of revolution select Volume of revolution. If there are multiple non-implicit function equations in the graph's inventory use the drop-down list at the top of the dialogue box to select the desired equation. The starting and ending values of the independent variable are entered into the "arc start" and "arc stop" boxes respectively. Specify the axis of rotation by either selecting the x axis, the y axis, or by entering values for a, b, and c in the equation ax + by = c . Enter a value for the number of "subintervals" (n) and press the "volume" button to obtain the numerical result. To compute the surface area of a solid of revolution select Surface area of rev and follow the same procedure as for a volume of revolution, but press the "area" button to obtain the numerical result.

    To see a three dimensional solid of revolution select Revolve surface If there are multiple equations in the graph's inventory use the drop-down list at the top of the dialogue box to select the desired equation. The starting and ending values of the independent variable are entered into the "arc start" and "arc stop" boxes respectively. Specify the axis of rotation by either selecting the x axis, the y axis, or by entering values for a, b, and c in the equation ax + by = c . Press the button "see surface" to display a 3D perspective plot of the solid which you can rotate by using the left, right, down and up arrow keys.

     

  1. To reflect a curve about a given mirror line choose the Reflect option. The mirror line can be specified as the x axis, the y axis, the line y = x (this generates the inverse relation to the original curve) or any line of the form ax + by = c . After designating the mirror line, press the reflect button to perform the reflection.



The Two pull down menu lists four procedures that Winplot can do to two functions at a time.

  1. To find the points of intersection of two curves select the Intersections dialogue box. If there are more than two functions of this type you choose which two to use by using the drop-down lists. The "next intersection" button is used to find the next meeting point. Only points of intersection within the graphing window are found. The results are added to the data text file and may be viewed by selecting Data/Inspect from the Misc pull down menu. The intersection coordinates may be saved to variables and the "mark point" button can be used to highlight them on the graph.
  1. To add, subtract, multiply, divide, exponentiate or compose two functions select the Combinations dialogue box.
  1. To numerically approximate the difference of two functions select the Integrate dialogue box.

From the Equa menu Shading can be used to shade the area between two curves over a specified domain.

  1. To visualize 3D solids that have a base in the x, y plane and specified cross section select Sections. The base is the region between two curves each defined by an equation of the form y = f(x). The specified cross sections are generated by planes perpendicular to the x axis and may be chosen from a variety of geometric forms. Pressing the "see solid" button displays a 3D perspective plot which you can rotate by using the left, right, down and up arrow keys. The "volume=" button displays a numerical approximation to the definite integral representing the solid's volume.

Saving, printing and copying a graph

To save all of the equations, windows and related settings associated with a Winplot graph choose Save or Save As from the File menu. Type in the desired file name and click the "Save" button. If you want to save to a particular location, insert the path before the file name or use the pull-down list to access the desired device or directory. For example, to save a Winplot graph called "mygraph" to the floppy drive, type a:\mygraph in the File name: box. To retrieve a graph, choose Open from the File menu and give the desired path and filename of the previously saved Winplot file.

To print your graph:

  1. Choose Format from the File menu. This opens the printing format dialogue box. Here you specify the width of the graph to be printed and the vertical and horizontal offsets of the graph from the upper left-hand corner of the page. A width of 15 cm or 16 cm (5.9 in to 6.3 in) will fill up most of the width of a page. Click on the "frame image" check box if you want a rectangular frame around your graph. Click on the "color printer" check box if you have a color printer and want your graph to print in color.
  1. Choose Print from the File menu and press the "OK" button.
To paste a copy of a graph into another Window's application (such as a word processor or a graphics editor) choose "Copy to clipboard" from the File menu for applications that recognize the Windows "metafile" format. If this doesn't work, try "Bitmap to clipboard" from the File menu.
 
 

Graphs in 3D

Many of the features (the inventory dialogue box, the conventions for entering formulas and function names, the insertion of text, the use and control of parameters, the printing and saving of files, etc.) of Winplot's 2D graph module are implemented identically in the 3D graph module.

Entering equations and viewing surfaces

There are six primary ways of generating a 3D graph by entering an equation.

1) Explicit        Graphs a surface with z as an explicit function of x and y.
2) Parametric   Graphs a parametric surface of the form x = f (t, u),  y = g(t, u), and z = h(t, u), for explicit functions f , g and
                        h and a pair of parameters t and u .
3) Implicit        Graphs an implicit surface defined by the relationship expressed in the equation f(x, y, z) = constant.
4) Cylindrical   Graphs a surface with z as an explicit function of r and t , where r and t are polar coordinates.
5) Spherical     Graphs a surface with r (the spherical coordinate "Rho" or distance from the origin) as an explicit function of t
                        (the spherical coordinate theta, the azimuthal angle about the z axis) and u (the polar coordinate phi measured
                       from the z axis).
6) Curve         Graphs a parametric space curve of the form x = f (t), y = g(t), and z = h(t), for explicit functions f , g and h of
                       a single parameter t .

For surfaces generated by explicit functions it is probably best to use the "quick" drawing mode. This provides greater speed and ease in both generating and modifying surfaces. This option is on when the item Fast draw mode is checked in the View menu. For this reason it is probably more convenient to generate a surface with several explicit formulas rather than use a single implicit formula. For example, to generate the surface of the elliptic hyperboloid of two sheets described by the equation

,

solve for z and graph the following explicit functions:

.

Using these two Cylindrical equations generated the figure shown below.

Below is a bitmap copy of an Explicit surface generated using the fast draw option.

This same surface could be generated using a parameter in the decaying exponential as shown below. This parameter could then be controlled from the Anim menu to give a dynamic "breathing" surface!

Clicking Explicit from the Equa menu results in the following dialogue box.

Define the domain of the function by entering the extreme values for x and y. Only rectangular domains are allowed. This means a surface whose base is a region in the x, y plane defined by non-constant boundary functions of y in terms of x or x in terms of y can not be generated by this option. The methods for drawing the graph make use of the x and y division values. Increasing these gives a "smoother" graph but at the expense of greater computation time. The upper and lower sides of the surface will be shaded differently if the "shade" box is checked and if different colors are picked via the "col 1" and "col 2" boxes. Click on the "ok" button to see the graph.

In the following example the value of the parameter A was set at 0.24 by the scroll bar selected from the Anim menu.

To compute a numerical approximation to the double integral of f (x, y) over its rectangular domain choose Integrate from the One pull down menu.

The procedure for entering a surface z = f(r, t) in Cylindrical mode is essentially the same as for entering a surface with
z = f(x, y) . Entering a surface in Spherical mode is similar, but one needs to remember that it is the radial distance rather than the z coordinate which is given as a function of the spherical angles.

From the Inventory menu you can select "levels" which generates level curves for a given surface.

Enter a value into the box labeled "level values for z". For surfaces defined by the Explicit option the surface from z = "low" to z = "high" is sliced by this number of equally spaced planes parallel to the x, y plane. The intersections between each plane and the surface are then projected onto the x, y plane resulting in a given level curve. The location of these intersection points involves a numerical search. The quickest method is press the "auto" button and let the program search over all the levels in succession. For some of the level curves this may miss some points. If this happens you can refine the search by specifying the value of z in the "curr" box and press the "search curr" button. Because the program does not know when to stop, it is necessary to press "Q" to quit. To obtain a contour plot press the button "see all". This draws all of the level curves in a single 2D graph. To superimpose the level surfaces onto the 3D perspective surface plot uncheck the "Fast draw all" item in the View menu.

Graphs of implicit surfaces f(x, y, z) = constant are generated by using the "levels" menu. Choose a level by selecting one of the three coordinate variables x, y or z and entering a value for that variable into the "current level" box. To see the level curves in the 2D window, click the "search level" button. The program will search for starting points and then draw the resulting contours, until you press "Q" to quit that level. The level values can be chosen individually, or you can set up a uniform selection by entering a count into the "level values for" box and extreme values into the "low" and "high" boxes. Click the "next level" or "previous level" buttons to enter a value into the "curr" box for you. As before, clicking the "auto" button will automatically search each level.

As the level curves are found, they are displayed in a 2D window and entered into a drop-down inventory box. You can delete unwanted curves from this list. When the level-curve inventory for the surface is satisfactory, click the "keep changes" button to exit the dialog and see the levels in 3D perspective. If you click the "discard changes" button, all editing is ignored and no changes are made to the 3D figure.

As an example of entering a space curve consider the periodic trajectory defined as follows:

The parameter t could be interpreted as the polar angle theta. For this curve to "retrace" itself the domain of t needs an interval of length 4*pi. Sample inputs and outputs for this trajectory are shown below.


 

Some View Menu Options

As in Winplot's 2D graph module the View menu in the 3D module allows you to alter the appearance of graphs.

Checking Axes displays x, y, and z axes in the 3D plots. This is useful in establishing the orientation of the figure.

Rotate changes orientation. In particular, adjusting Up (use the up arrow key), Down (use the down arrow key), Turn (use the right arrow key), or Back (use the left arrow key) rotates the figure. Pressing the up arrow decreases the polar angle, "phi", by a fixed amount (the default is 6 degrees, but this can be reset in the Angle dialogue box). Pressing the right arrow key increases the azimuth, "theta", about the z axis by this same fixed amount.

Zoom/Out (use the Page Down key) shrinks the size of the figure by a fixed ratio (the default is 1.1 but this can be changed in the Zoom/Factor dialogue box). Similarly, Zoom/In (use the Page Up key) increases the size of the figure.
 
 

Plotting Phase Space Trajectories of Differential Equations



The Differential option from Winplot's 2-Dim Equa menu can generate graphs of slope fields and solutions for first order differential equations (the dy/dx option) or graphs in two dimensional phase space for a system of first order differential equations in two dimensions (the dy/dt option). Such systems are often referred to as dynamical systems and even an introduction to such problems is outside the scope of this tutorial. The discussion here will be limited to using the dy/dt module to plot the direction field of a vector field or to generate phase space trajectories for second order ordinary differential equations.

Plotting the direction field of a vector fields in 2D

The 2-dim Differential dy/dt module is designed to generate plots in x, y space of solutions to the dynamical system defined by the first order system:

If the functions F and G are independent of the parameter t the system is called autonomous. Suppose this is the case, then from the chain rule

Thus, the solutions in x, y space follow the direction of the vector field

To plot the direction field of a vector field use the 2-dim Differential dy/dt module as follows:

  1. From the Equa pull down menu choose the Differential dy/dt option.
  2. In the dialogue box shown below enter the equation for F(x, y) in the box for x prime.
  3. Enter the equation for G(x, y) in the box for y prime.
  4. Click on the "vectors" radio button.
  5. Adjust the color, the length of the vectors in "lengths (pct of screen width)", and the number of "horizontal rows" to give a reasonable graph within the graph window.
  6. If necessary, change the graph window parameters in the View menu

This is illustrated in the following two examples.

Example 1: Plot the direction field of the radial vector field

Example 2: Plot the direction field of the circulating vector field

Generating Phase Space Trajectories for Second Order Differential Equations

The use of phase space to describe solutions of second order ordinary differential equations is introduced in the following web site: http://faculty.madisoncollege.edu/alehnen/webphase/sld001.htm.

To plot the phase space trajectories of a second order ordinary differential equation use the 2-dim Differential dy/dt module as follows:

  1. From the Equa pull down menu choose the Differential dy/dt . option.
  2. In the dialogue box enter y in the box for x prime.
  3. Enter the equation for the second derivative of x in the box for y prime.
  4. Click on the "vectors" option. This plots the unit tangent field of the trajectories.
  5. Adjust the color, the length of the vectors in "lengths (pct of screen width)", and the number of "horizontal rows" to give a reasonable graph within the graph window.
  6. If necessary, change the graph window parameters in the View menu
  7. From the One menu select dy/dt trajectory to access the IVPs dialogue box to superimpose on the unit tangent field particular solutions corresponding to various initial conditions. The radio button "fwd" runs these solutions forward in the parameter t. The radio button "rev" runs these solutions backward in the parameter t. The radio button "both" runs these solutions both forward and backward. Pressing either the "draw" or "watch" buttons graphs the solution in phase space. If you choose "watch", you can slow the drawing of the solution by entering a positive number in the "delay" box. Additional solutions are entered by entering additional initial conditions. Each new solution is added to the inventory list. Remove any unwanted solutions by highlighting them and clicking the "delete" button.
This is illustrated below for the second order differential equation describing a simple pendulum.




These notes were authored by Al Lehnen, a math instructor at Madison Area Technical College in Madison, Wisconsin.
I welcome any comments and suggestions. Please feel free to E mail me at alehnen@madisoncollege.edu or write to me at the following address.

Madison Area Technical College
3550 Anderson Street
Madison, WI 53704

My special thanks to high school math instructor Beverly Leacock of Lewisville, Texas who has updated and modified these notes for her colleagues and students and whose suggestions have helped me to improve this document.