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The Maplet Interface for ODE Analysis

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A Maplet Interface for ODE Analysis
by Maplesoft

Maple has long been famous for the power of its ODE solvers, but the wealth of available options and algorithms can be overwhelming to new users.  Maple 9 provides a Maplet interface to its ODE solvers and plotters called the Interactive ODE Analyzer .  Users can now bring Maple's advanced options to bear on ODE systems without having to learn the syntax of dsolve  and odeplot .

Exploring a numeric solution

In this example, we use the Interactive ODE Analyzer Maplet to explore the chaotic Lorenz attractor numerically.

To invoke the Interactive ODE Analyzer, use the new interactive  option to dsolve . You can call it on a predefined ODE system (as done below) or with no arguments and enter the information through the maplet.  You can open separate windows to edit the DE's, conditions and parameters by clicking the corresponding  buttons.

>    L  := diff(x(t),t) = a*(y(t)-x(t)),
     diff(y(t),t) = b*x(t) - y(t) -x(t)*z(t),
     diff(z(t),t) = x(t)*y(t) - c*z(t);

L := diff(x(t),t) = a*(y(t)-x(t)), diff(y(t),t) = b*x(t)-y(t)-x(t)*z(t), diff(z(t),t) = x(t)*y(t)-c*z(t)

>    dsolve[interactive]( { L } );

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In the Edit Parameters window, you can change the parameter values.

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In the Edit Conditions  window, you can edit, add or delete initial and boundary conditions.  Click Add  to add the condition to the list of existing conditions.  Click Done  to return to the main window.

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When all parameters and conditions are specified, you can solve the system numerically by clicking the Solve Numerically  button.

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The Solve Numerically window contains most of the functionality of dsolve/numeric .  Select the solving algorithm in the left half of the window.  Some numeric ODE algorithms have submethods and parameters, which you can specify through pull-down menus.

To see solution values at a particular value of the independent variable, enter the value in the first text region on the right and click
Solve .

To plot the solution, click
Plot , or first choose plot options by clicking Plot Options .

If the
Show Maple commands  checkbox is checked, the window displays the Maple commands that would be needed to reproduce the output in the worksheet.  This is a very handy feature for learning the command syntax of dsolve  and odeplot .

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The Plot Options window contains most of the options of   plots[odeplot] .  By picking Axis 1 , Axis 2  and Fcn  appropriately, you can create plots of any dependent function (or its derivatives) vs. the independent variable, or phase portraits among the dependent functions (or their derivatives).  

You can include multiple curves in the same plot, each with different dependent functions, conditions and parameters.

Done  to return to the Solve Numerically window.

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If you generate a 3-D plot, you can rotate it in real time with the mouse, just like you can to 3-D plots in the worksheet. This is a new Maplets feature in Maple 9.

When you close the Maplet (
Quit  button), you can take any of the maplet's results back with you to the worksheeet, including the plot, the numeric procedure returned by dsolve , or the Maple commands used to generate the outputs. Select this option from the On Quit, Return  pull-down menu.

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Exploring a symbolic solution

>    ode  := x^(n-1)*diff(y(x),x)^n-n*x*diff(y(x),x)+y(x);

ode := x^(n-1)*diff(y(x),x)^n-n*x*diff(y(x),x)+y(x)

>    dsolve[interactive]( { ode } );

Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct

y(x) = 1/exp(-I*Pi*n/(n-1))*(-1/exp(-I*Pi*n/(n-1)))^(-n)-exp(-I*Pi*n/(n-1))*(-x/exp(-I*Pi*n/(n-1)))^(1/n)*n


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Here we get a symbolic solution.  To explore an IVP for a particular value of n, we click the Back  button to return to the main window and set n through the Edit Parameters  window.

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Finally, we tell the maplet to return the solution and quit.

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