After a recent webinar in which I interactively solved the initial-value problem (IVP)

with the ODE Analyzer Assistant, I received an email asking how one could visualize the solution if one or two of the coefficients were parameters. If the coefficient of  were the parameter , could a solution  be graphed as a surface on which the specific solution  were superimposed as a space curve whose projection onto the -plane would be the solution of the IVP with . In particular, could the value of  be controlled by a slider.

If the coefficients of  and  were  and  respectively, could Maple simultaneously send a coordinate pair  to a solver which then graphed the solution of the IVP in which . In particular, could Maple reproduce the "2D slider" available in such software programs as Geometer's Sketchpad?

Below, both questions are answered in the affirmative.

The surface in Figure 1 is the solution  of the IVP

The slider controls the value of the parameter , and as it is moved to the value , the solution  is graphed on the surface as a space curve. The code by which this is accomplished can be seen by right-clicking on the slider and selecting "Edit Value Changed Action..." For this figure, the IVP is solved analytically for the function , and as the slider varies the value of , the space curve  is appended to the graph of the surface.

This answers the first question.

The curve in Figure 2 is the solution of the IVP

in which the values of the parameters  and  are controlled by separate sliders.

The code behind each slider is the same, a technique that is naive at best. A more efficient strategy would be to write the code as one or more functions in a start-up code region, and then make appropriate function calls behind the sliders.

Compare Figure 2 with Table 32.1 in Gem 31, where the Explore command is used to generate the solution of this IVP, again with two separate sliders to control the parameters  and .

The graph on the left in Figure 3 shows a portion of the -plane and the curve . The characteristic equation for the differential equation

is , with solutions . Hence, points  on the curve  correspond to systems that are critically damped; points in the red region (where  correspond to systems that are underdamped; and points in the yellow region (where ) correspond to systems that are overdamped.

Click and drag on the left-hand graph to select parameter pairs ; the corresponding solutions of the IVP

are drawn in the right-hand graph. (The default manipulator for the left-hand graph has been set to "Click and Drag". If, for any reason this default changes, use the Context Menu for the graph to re-select Manipulator/Click and Drag.)