Introduction to Systems of Ordinary Differential Equations
 

Systems of ODEs generates numerical solutions of systems of up to 6 first order ordinary differential equations containing parameters. These solutions can be tabulated and plotted. A user supplied function can also be plotted.

The independent variable is "t", and the dependent variables are "u", "v", "w", "x", "y", and "z". The allowable parameters are "a" though "r", excluding "e" (which has its usual numerical meaning).

 
 Creating and Loading Systems of ODEs
 

When you first run the program the Main Window is opened, and you will see that essentially the only commands available, are "New", "Open System", and "Open Environment" on the toolbar, and "New System", "Open System", and "Open Environment" under the "File" menu.

Selecting "Open System" or "Open Environment" allows you to load System files (ending with "syw", "en1, "en2", and "en4") that have been previously created. If you use "Open System" or "Open Environment", then the current ODEs are overwritten by the loaded ODEs, and lost.

To create a system, complete the appropriate boxes alongside "du/dt", "dv/dt", ... , "dz/dt", along with the initial values, "u(t0)", "v(t0)", ... , z(t0)". The initial time is "t0" and the final time is "t1". If "t1" is less than "t0" then solutions are generated backwards in time.

The ODEs may contain

• The independent variable "t".
• Any of the dependent variables "u", "v", "w", "x", "y", "z".
• Any of the allowed parameters "a" though "r", excluding "e".
• The constants "e" and "pi".
• The standard mathematical functions and operations that the program recognizes.

The initial values may contain

• Any valid number.
• The constants "e" and "pi".
• Any of the above two combined lawfully with the operators +, -, *, /, and ^ (exponentiation).

An optional function f(t) can also be defined. Usually this would be an exact solution of the system of ODEs. The function may contain

• The independent variable "t".
• Any of the allowed parameters "a" though "r", excluding "e".
• The constants "e" and "pi".
• The standard mathematical functions and operations that the program recognizes.

The functions that the program recognizes can be seen by clicking on "View Recognized Functions and Operators" button, which displays the "Recognized Functions" window. A more detailed summary can be found here.

A single-click opens a tooltip, with an explanation of the function. There are three different ways to cut-and-paste a function to an ODE box.

• Single-click on the function, right-click to open a drop-down menu, and select "Cut".
• Single-click on the function, and then click "Copy".
• Double-click on the function.

After the ODEs (and optional function) have been created, pressing the "Accept" button checks the validity and consistency of these equations. The program automatically removes all spaces and converts all characters to lower case.

If any parameters are used in the ODEs then they are exposed under "Parameters". The parameter values, like the initial values, may contain

• Any valid number.
• The constants "e" and "pi".
• Any of the above two combined lawfully with the operators +, -, *, /, and ^ (exponentiation).

The step-size "s", which defaults to 0.01, can be changed. This is the number used by all the numerical and plotting techniques.

The following toolbar commands are now active:

• "Save System", which saves the existing ODEs to a Systems file with extension "syw".
• "Edit", which allows you to edit the existing ODEs.
• "New", deletes the existing ODEs, and starts afresh.

• The "Numerical" button. Clicking this automatically opens the Numerics Window.
• The "Plot 1 Window" button. Clicking this automatically opens the Plot 1 Window, which plots numerical solutions in one graphics window.
• The "Plot 2 Windows" button, if there is more than one ODE. Clicking this automatically opens the Plot 2 Window, which plots numerical solutions in two graphics windows.
• The "Plot 4 Windows" button, if there is more than one ODE. Clicking this automatically opens the Plot 4 Window, which plots numerical solutions in four graphics windows.