Instructions for using the Differential Equations Applet

Go to the slope field applet for dy/dx

Go to the slope field applet for dx/dt

Go to the direction field applet

Entering/editing differential equations
Mathematical functions reference
Mathematical operators reference
Setting variable/parameter ranges
Zooming into a region
Grid controls
Display controls
Adding initial conditions
Plotting solutions
Choosing the axes of the plot
Viewing solution tables
Printing from the Applet

Entering differential equations

Enter the right hand side of the equation into the area after
```
eqn #k: dy/dx= 
```
The number k is used to identify the equation number in error messages, so that possible problems can be traced to a particular equation. In the default mode, the applet has only one equation. Equation #0, if present, contains initialization code. This code is evaluated only once after its change. It is typically used to define constants and functions.
Any expression in variables x and y is allowed, using operators +,-,* and /, and mathematical functions from the following list: sin, cos, tan, arctan, arcsin, exp, ln, sqrt, abs, min, max (these last two are binary). Mathematical constants "E", "PI" and "Infinity" are also available. Examples of valid expressions:

sin(x)/y
y^2*sin(x)
min(x,y)
abs(x^2-y)
sin(PI*y)
exp(y)
E^y

The equation is entered into the Applet by hitting Return or by pressing the "Submit All" button. Immediately after submission, the slope field and solutions are going to be plotted.

Function reference

The following functions are available for use in expressions:

sin(x): Returns sine of x.
cos(x): Returns cosine of x.
tan(x): Returns tangent of x.
artcan(x): Returns inverse tangent of x.
arcsin(x): Returns inverse sine of x.
abs(x): Returns absolute value of x, i.e. x for x>=0 and -x for x< 0.
exp(x): Returns the exponential of x.
ln(x): Returns the natural logarithm of x.
sqrt(x): Returns the square root of x for x>=0. If x< 0
min(x, y): Returns x if x< y and y otherwise.
max(x, y): Returns y if x< y and x otherwise.
step(x): Returns 0 for x < 0 and 1 for x >= 0.
sgn(x): Returns -1 for x < 0, 0 if x=0 and 1 for x > 0.
iffun(x, y, z): Returns y for x is not equal to 0.0, z otherwise. Typically x is written as a logical expression (see operator reference). Note: This is a true function, i.e. it evaluates all its arguments (x, y, and z) before returning its value.

Mathematical constant reference

E: The base of natural logarithm.
PI: The circumference of a circle of diameter 1.
Infinity: Positive infinity.

Operator reference

In addition to the ordinary arithmetical operators +, -, *, /, the following operators are available for use in expressions:

x^y

Has a value of x raised to power y.

x < y

Has a value of 1.0 if x is less than y and 0.0 otherwise.

x <= y

Has a value of 1.0 if x is less or equal to y and 0.0 otherwise.

x > y

Has a value of 1.0 if x is greater than y and 0.0 otherwise.

x >= y

Has a value of 1.0 if x is greater or equal to y and 0.0 otherwise.

x == y

Has a value of 1.0 if x is equal to y and 0.0 otherwise.

x != y

Has a value of 1.0 if x is not equal to y and 0.0 otherwise.

x && y

Has a value of 1.0 if x and y are different from 0.0. Otherwise it has a value of 0.0.

x || y

Has a value of 1.0 if either x or y is different from 0.0. Otherwise it has a value of 0.0.

x; y

Has a value of y. The expression x is evaluated for its side effects. Typically x is an assignment, as in

k=7; k^3-k^2-1

Note: In logical expressions the value of 0.0 is equivalent to false and any value not equal to 0.0 is considered true. Logical operators return 1.0 when the logical expression is true. This resembles the convention used by languages like C and Java.

Plotting solutions for selected initial conditions

Click your mouse in the canvas area to select an initial condition. A solution passing through the indicated initial condition will be plotted. You can also select the initial condition precisely by typing in values of x and y. This method is described in this section.

Note: If there are more variables than 2, only two variables will be set based on the position of the mouse. The remaining variables will be set to 0.

Choosing abscissa and ordinate of the plot

If the number of variables is greater than 2 (including the independent variable) then you may select the variables that will be used as the abscissa and ordinate of the plot. This is done by pressing two choice buttons placed along the left and bottom edge of the plot and selecting one of the list of all variables.

Viewing solution tables

The applet tabulates numerical solutions of the differential equations. You can view tabulated values for each initial conditions in a window by clicking on the button labelled "Show table". A window entitled "Solution table" will pop up with all numerically generated solutions, in a form resembling tables in standard textbooks on numerical methods. One can use Cut-And-Paste to copy these tables into a text editor or word processor. The applet generates a solution table every time it recalculates the solution. Old solution tables can be erased by pressing the "Clear" button of the table window.

Setting ranges of variables

These are entered into areas labeled as "Min. x", "Max. x", "Min. y" and "Max. y" etc. They are not used in calculations, until they are entered by either pressing the Return key, or by pressing the "Submit All" button.

Setting the sizes of the grid

The sizes of the grid in the x- and y- and other directions are entered in areas marked as "Num. of segs" . They become effective upon either hitting the Return key (one item at a time) or by pressing the "Submit All" button, which updates all input related to entering the formula for dy/dx, the ranges and the number of grid intervals.

Zooming into a region

The right mouse button, on machines with 2 and 3 mouse buttons, or pressing the mouse button while the META key is depressed on machines with a single mouse button, will initiate a selection of a rectangle in the canvas. By dragging the mouse, the rectangular region is selected. When the mouse button is released, the range of the two coordinates is reduced to the selected rectangle. The zoom operation can be undone by pressing middle mouse button on system with three mouse buttons (or where the "wheel" can be used as middle mouse button) or by pressing the ALT key simultaneously with the mouse button.

Display controls

Slopes

If checked, slopes are displayed.

Solutions

If checked, solutions with selected initial conditions are displayed.

Init. Conditions

If checked, every initial condition is marked and labeled.

Euler/ModifiedEuler/Midpoint/Runge-Kutta/RK4/Runge-Kutta-Fehlberg/RKF

Selects the algorithm used in calculating solutions.

Step

Displays and sets the integration step. Integration step is set automatically in the following circumstances:

the number of intervals (fields following "Num. of intervals") is set;
the "Submit All" button is pressed.

The automatic setting is calculated by dividing the range of the independent variable by the number of intervals. The automatic setting of the step can be overridden, though, by filling in the value of the step and hitting the Return key.

Adding initial conditions

The Applet maintains a list of initial conditions whose solutions will be plotted. You can:

Add an initial condition by setting the fields following the label "Add init. cond.:". There is a field corresponding to each variable.
Remove all initial conditions and start anew by pressing the "Clear All" button.
Initial conditions can be added by clicking them with the mouse.

Displaying tables of solutions

All solutions displayed by the applet can be tabulated by pressing the "Show table" button in the initial condition entry group of buttons. Tables are displayed in a separate window. The table can be edited or copied into another application (e.g. text editor). This is achieved by first selecting the data to be copied (typically with a mouse) and then with the usual Control-C (Cut) and Control-V (Paste) key combination, or other system-specific Cut/Paste keys. The table data can also be printed directly from the applet by selecting "Pring" from the "File" menu. Further necessary information on printing from the applet is contained in this section.

Printing from the Applet

The best general advice to avoid problems with printing (due to various bugs in major browsers and Java implementations) is to accept the certificate. The dialog to accept the certificate should pop up automatically when the page containing the applet is loaded for the first time.

The page containing the Applet can be printed with all major browsers. However, there are some browser peculiarities. The problem is especially aggravating with Microsoft Internet Explorer, because the applet window is reset to its original state (i.e, your data is erased ):- ) before printing. Thus, in order to print, manipulating the security settings of the browser is required, or downloading a certificate file, which certifies the applet's authenticity. Unfortunately, the number of various browser version combinations and strategies is too numerous to discuss in detail. The included HTML files should take care of making the choice for various Netscape and Internet Explorer versions. The good news is that for versions 6.0+ of the two major browsers the only required action to print is accepting the certificate. Other strategies are explained below.

Microsoft Internet Explorer 6.0-

The default security settings do not allow unsigned applets to print. In order to print from the applet in Internet Explorer 5.5 you should allow the applet Web server (i.e. the Web server from which you downloaded this applet) to print. This is done as follows:

Select the "Internet Options/Security" menu.
One needs to add this Web server to the list of trusted sites. Click on "Trusted sites" image. Add this server to the list of trusted servers. Close the dialog after doing so.
Again, select "Trusted sites" icon on the "Internet Options/Security" menu. Click on the "Custom level" button. The "Security Settings" dialog will pop up. Following "Virtual Machine/Java settings" check the radio button labelled "Custom". Subsequently, press button "Java Custom Settings" at the bottom. A new dialog "Trusted sites" will pop up. Find "Printing" and enable it.

If you simply use the browsers "Print" button to print the version of the Applet embedded into the Web page, Internet Explorer will initialize the Applet before printing, and thus any changes, initial conditions, etc. will be lost. Therefore you must use the built-in "Print" button of the Applet, or the "Print" from the "File" menu of the Applets's window if running the Applet in a separate frame. You must also accept the security certificate when the Applet first loads.

Netscape 6.0 and Mozilla

The applet should print with these browsers after accepting the certificate. The dialogue to do so will pop up automatically.

Appletviewers

You can also print the Applet using Sun's appletviewer. Microsoft viewer JView does not allow printing.

Errors

These are typically generated by typing in an expression (see section on entering differential equations) which does not conform to the syntax rules of the Applet. The most recent error message appears in the box labeled "Last error". It will hopefully identify the problem by pointing to the column in which the first error occurs. If not, by pressing the button labelled "Show All Errors" all error messages can be examined and used to correct the problem.

Enjoy,

Marek Rychlik (rychlik@u.arizona.edu)

Instructions for using the Differential Equations Applet

Contents

Microsoft Internet Explorer 6.0-

Netscape 6.0 and Mozilla

Appletviewers

Errors