Using the Interactive ODE Analyzer Assistant

The ODE Analyzer Assistant provides an interactive application for solving ODEs symbolically or numerically, viewing plots of the solution, and seeing the Maple commands used to generate the solution.


To launch the ODE Analyzer Assistant, do one of the following:

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From the Tools menu, select Tutors, Differential Equations, and then ODE Analyzer.

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Write the differential equation in a Maple worksheet, and then from the Context Panel, select Solve DE Interactively.

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Enter the command dsolve[interactive](); at the Maple prompt.


Note: With the both the second and third methods you can enter the differential equation in a Maple worksheet, which may be easier. See the section Alternative Entry of the Problem.


Main Window


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The Interactive ODE Analyzer Assistant main window displays the input system in three parts.


 Differential equations of the system


 Initial or boundary conditions in the system


 System parameters (unknown constants)

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To modify input, click the Edit button directly beneath the equations, conditions, or parameters display.


Note: The initial conditions or parameters can be edited only if a differential equation has been entered.

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To solve the given system numerically or symbolically, click the Solve Numerically or Solve Symbolically buttons respectively. The option specification window for the numeric or symbolic solutions opens.

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To classify the given system click the Classify button, if available. When available, the Classify button uses DEtools[odeadvisor] to classify the ODE.

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The Quit button exits the assistant.


Edit Equations


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To access the Edit Differential Equations window, click the Edit button directly below the Differential Equations group box in the Main window.

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There are two primary areas on this window.


This area is used to add a new equation to the existing system. Enter the ODE in the input area, and click the Add button. The input equation is checked for validity and syntax at this point, and if valid, it is placed in the Edit Equations area below.


Note: The input of the Add Equation area is ignored on exit. If the equation is to be retained, then it must be added to the system first, by clicking the Add button.


This area is used to delete or modify the equations of the system. Only 4 equations are in view at one time. If more than 4 equations are present, these can be viewed by using the navigation buttons at the bottom of the Edit Equations area (which appear only when there are more than 4 equations in the system).


The Delete button deletes the equation to the left of the button, shuffling all equations below it.


The Edit button opens an Edit Equation window, which can be used to modify the corresponding equation.

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The Done button accepts all additions, deletions, or changes made to the equations, then returns to the Main window.

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The Cancel button rejects all changes made, and then returns to the Main window.



Edit Equation


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To access the Edit Equation window, click the Edit button to the right of the equation to be edited in the Edit Differential Equations window.


In this window, you can modify the equation, and click OK to accept changes made to the equation. Clicking Cancel rejects those changes. In either case, you are returned to the Edit Equations window.


Note: If the modified equation is invalid, you will be unable to leave this window until the equation is corrected, or the Cancel button is clicked.



Edit Conditions


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To access the Edit Conditions window, click the Edit button directly below the Conditions group box in the Main window.

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There are two primary areas on this window.


This area is used to add a new condition (initial or boundary) to the existing system. The Add Condition area has a dropdown box for selecting the dependent variable or derivative, and an area for entering the point at which the condition should be applied and the value at that point. When clicking the Add button, the condition is checked for validity and syntax, and if valid, it is placed in the Edit Conditions area below.


Note: The input of the Add Condition area is ignored on exit of the window. If the condition is to be retained, then it must be added to the system first, by clicking the Add button.


This area is used to delete or modify the conditions of the system. Only 5 conditions are in view at one time. If more than 5 conditions are present, these can be viewed by using the navigation buttons at the bottom of the Edit Conditions area (which appear only when there are more than 5 conditions in the system).


The Delete button deletes the condition to the left of the button, shuffling all conditions below it.


The Edit button opens an Edit Condition window, which can be used to modify the corresponding condition.

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The Done button accepts all additions, deletions, or changes made to the conditions, then returns to the Main window.

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The Cancel button rejects all changes made, and then returns to the Main window.



Edit Condition


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To access the Edit Condition window, click the Edit button to the right of the condition to be edited in the Edit Conditions window.


In this window, you can modify the condition, and click OK to accept changes made to the condition. Clicking Cancel rejects those changes. In either case, you are return to the Edit Conditions window.


Note: If the modified condition is invalid, you will be unable to leave this window until the equation is corrected, or the Cancel button is pressed.



Edit Parameters


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To access the Edit Parameters window, click the Edit button directly below the Parameters group box in the Main window.

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This window can be used to set or modify the values of any parameters present in the equations or conditions of the system.

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Clicking OK accepts changes made to the parameter values, while clicking Cancel rejects those changes, in both cases returning to the Main window.




Alternative Entry of the Problem


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As an alternative to entering the problem through the assistant, you can enter the differential equation in a Maple worksheet.

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For example, to launch the assistant prepopulated with the example used on this page, execute this Maple expression:

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{diff(x(t),t)=y(t)+2*x(t),diff(y(t),t,t)+x(t)=0,x(0)=1,y(0)=0,D(y)(0)=1};


And then, from the Context Panel for the output, select Solve DE Interactively

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Equivalently, you can launch the Interactive ODE Analyzer Assistant with a problem by using the command dsolve[interactive]:

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dsolve[interactive]({diff(x(t),t)=y(t)+2*x(t),diff(y(t),t,t)+x(t)=0,x(0)=1,y(0)=0,D(y)(0)=1});



Solve Numerically


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To access the Solve Numerically window, click the Solve Numerically button of the Main window.

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In this window, you can specify options for obtaining the numerical solution of the input system that display the relevant output (solution values or plots). Note: This window can only be reached if the input system is fully determined, that is, it is possible to obtain a unique numerical solution for the input, as all relevant conditions are specified, and all parameters have numerical values.


The Parameters area of this window allows you to specify the method, the error tolerances (when appropriate), and some of the methodspecific parameters used to obtain the numerical solution.


For initial value problems (IVP), a total of 8 major methods are available (some including submethods), while for boundary value problems (BVP) only one method is available. The radio buttons for all applicable methods are accessible, while those for methods that cannot be used for the ODE system are not.


Note: Not all options available for use with these methods are accessible from the interface, but those most commonly used are.


For more information on methodspecific options and submethods, click the the Help button at the bottom right of the window. The Help Navigator window opens. Also, see the relevant help pages, specifically, rkf45, dverk78, gear, rosenbrock, lsode, numeric,bvp, taylorseries, and classical.


The Output area provides solution values at a point (which can be changed as desired), to display a plot of the solution (options can be changed in the Plot Options window), and also displays the relevant Maple commands needed to accomplish these tasks (making this useful as a learning tool).


To obtain solution values, change the value in the Show function values at field to the desired solution point, then click the Solve button to the right. This produces the solution values at the requested point in the output, below the value of the requested point.


To change the requested point, click the Solve button. Solution values for the new point are provided.


The solution plot is displayed below the function values. If no specific plotting options have been specified via the Plot Options window (accessed by using the Plot Options button), then clicking Plot produces a plot of the numerical solution based on the default values of odeplot. By default, only the first solution component is plotted, as a function of the independent variable of the problem.


Example To adjust the plot settings to show plots of $x\left(t\right)$, $y\left(t\right)$, and $y\prime \left(t\right)$ in the example used here, the independent variable range was set to 5..1 and under Functions to plot, click the button under Copy to add two more curves, and specify their functions as $y$ and $y\prime$.


The Plot Options button opens the Plot Options window, which allows you to specify the function(s) to plot, the colors and style of the plot, the initial conditions for curves of the plot, the range of the independent variable used for the plot, and other plotspecific options.


The area below the plot region is used (optionally) to show the Maple commands needed to compute the solution value and/or produce the displayed plot. If the check box is selected, the commands are displayed, otherwise not.


The On Quit, Return dropdown box below the Parameters area allows you to specify what to return to the Maple session when Quit is clicked from this window. Options include nothing, the displayed plot, the computed numeric procedure, or the Maple commands needed to produce the solution values and/or the displayed plot.


The Clear button clears the solution values and the displayed plot in the output area.


The Help button opens a Help Navigator, which provides more information on the specific numerical methods and parameters.


The Main button closes the Solve Numerically window, returning to the Main window.


The Quit button closes the application, returning to Maple with the output specified by the On Quit, Return dropdown box. Note: If Plot is clicked, but there is no plot in the plot region, then nothing is returned. Similar comments apply for the Numerical Procedure and Maple Commands options.



Solve Symbolically


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To access the Solve Symbolically window, click the Solve Symbolically button in the Main window.

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In this window, you can specify options for obtaining the symbolic solution of the input system that display the relevant output (solution or plots). Note: This window can be reached for systems with arbitrary conditions or undetermined parameters, but plots are not possible until those unknowns are specified numerically.


In the Method area of this window, you can specify the method, and various methodspecific options. The unavailable options are grayed out for situations in which they cannot be used. For example, if the input is a single ODE, then the choice or restriction of methods is available for the Default method, but not for systems. As another example, the Formal series method only applies to a single linear homogeneous ODE; therefore, for any other ODE or system it is not available.


Note: Not all options available for use with these methods are accessible from the interface, only those most commonly used.


For more information on methodspecific options and submethods, click the Help button. The Help Navigator window opens. Also, refer to relevant help pages, specifically, dsolve, dsolve,system, dsolve,inttrans, dsolve,series, and dsolve,formal_series.


The Output area provides a symbolic solution, or displays a plot of a fully specified solution.


At the top of the output area is a region devoted to display of the symbolic solution, which can be obtained by specifying the solution method, then clicking the Solve button.


If the solution is large or complex, it is possible to display it alone in a larger window by clicking the Large Display button. For the example used in this help page, the following is displayed.


Below that is a plot region, used to display plots of fully specified solutions. To display a plot, click the Plot button. Note: In the event that the solution is not fully specified, clicking the Plot button opens the Plot Options window, from which it is possible to set the undefined parameters of the solution. If these are not set to numerical values, then no plot is displayed.


The Plot Options button opens the Plot Options window, which allows you to specify the function(s) to plot, the colors and style of the plot, any undetermined parameters in the symbolic solution, the range of the independent variable used for the plot, and any other plotspecific options (described in more detail in Plot Options below.)


The area below the Method area is used (optionally) to show the Maple commands needed to compute the solution value and/or produce the displayed plot. If the check box is selected, the commands are displayed, otherwise not.


The On Quit, Return dropdown box below the Maple commands display allows you to specify what to return to the Maple session when Quit is clicked from this window. Options include nothing, the displayed plot, the symbolic solution, or the list of Maple commands needed to produce the solution values and/or the displayed plot.


The Clear button clears the symbolic solution and the displayed plot in the output area.


The Help button opens Help Navigator, which provides more information on the specific numerical methods and parameters.


The Quit button closes the application, returning to Maple with the output specified by the On Quit, Return dropdown box. Note: If Plot is clicked, but there is no plot in the plot region, then nothing is returned. Similar comments apply for the Symbolic Solution and Maple Commands options.



Help Navigator


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To access the Solve Numerically Help window or the Solve Symbolically Help window, click the Help button on the Solve Numerically or Solve Symbolically windows respectively.

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This window provides basic highlevel information on the numeric or symbolic methods available in dsolve[interactive]().

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You can choose a topic from the Help for dropdown box. Once a topic is chosen, the information appears in the area below. button at the bottom of the window is enabled, and the format width of the displayed help page can be controlled as well.

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When finished with help, click the Dismiss button to close the window.



Plot Options


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This window controls the settings of the plot options, such as title, axes labels, 2D or 3D plotting, curves to plot, independent variable range, and numerous other options. The window contains three main areas.


The General Options area controls highlevel plot options, including whether the plot is 2D or 3D, what style of axes to use, the number of points in the plot, the title of the plot, and axes labels.


Only options that are valid for the current settings can be changed. For example, if Axes is set to none, then Axes Labels cannot be specified.

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Axes and Independent variable range


This area controls what part of the curve is displayed, and for 3D, the orientation of that display.


The Axes choices are restricted to the independent and dependent variables of the ODE or ODE system. Axis 1 represents the horizontal axis for 2D plots. Axis 1 and Axis 2 represent the x and y axes of the plot, respectively, for 3D plots.


The independent variable range specifies the portion of the ODE solution to be plotted.


The range choices (for example, Axis 1 range) allow you to restrict the viewing range for the plot, and are equivalent to the view option in plot and plot3d.


The 3D orientation choices allow you to specify the viewing angle for a 3D plot, and are described in plot3d,option.

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Functions to plot and corresponding conditions/parameters


This area controls what values are plotted on the dependent axis (the vertical axis for 2D plots, and the zaxis for 3D plots). By default, 4 curves can be plotted simultaneously, but this can be set from 1 to 16 by using the maxplots parameter to dsolve[interactive].


Each row in this area specifies a single curve to be plotted, and all curvespecific options to use for that curve. The function, color, and line style, and size can be changed, as well as some parameters of the solution itself. For example, for numerical solutions, the initial conditions/boundary conditions can be changed for a specific curve, allowing comparison of two solutions with different initial conditions. For symbolic solutions, free parameters in the solution of the conditions can be changed, allowing similar flexibility.


To add another curve to be plotted:

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Select a curve with similar characteristics, then click the (") button in the Copy column next to it. A duplicate of that curve is placed in the next available row, which can then be modified.


To remove curves, click the (X) button in the Delete column next to the corresponding row.

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The Done button closes the Plot Options window, accepting all changes.

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The Cancel button closes the Plot Options window rejecting all changes.






