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Plotting Three-Dimensional Differential Equations

This worksheet describes the DEplot3d routine, located in the DEtools package. Given a system of differential equations and a list of initial conditions, DEplot3d produces a three-dimensional representation of the solution curves of the system.

In order to access the routines in the DEtools package by their short names, the with command has been used.

$restart$

>	$with (DEtools) &colon;$

Options

The DEplot3d supports a wide variety of options. These options are summarized in the following sections.

iterations and stepsize

The option iterations allows the user to ask that steps of size stepsize be taken between output points. This gives greater accuracy in the output points, while reducing the actual number of output points that have to be stored.

>	$DE1 := \{\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x) = y (x) z (x), \frac{&DifferentialD;}{&DifferentialD; x} z (x) = y (x) x\}$

>	$DEplot3d (DE1, [y (x), z (x)], x = - 2 .. 2, [[y (0) = 1, D (y) (0) = 2, z (0) = 1]], y = - 4 .. 4, obsrange = true, stepsize = 0.05, iterations = 3, orientation = [- 124, 72])$

$DE1 ≔ \{\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x) = y (x) z (x), \frac{&DifferentialD;}{&DifferentialD; x} z (x) = y (x) x\}$

linecolor

The linecolor option works in a way similar to that of color, but it is applied to the solution curves.

plot[color] Name

>	$DE2 := \{\frac{&DifferentialD;}{&DifferentialD; t} x (t) = y (t), \frac{&DifferentialD;}{&DifferentialD; t} y (t) = - \sin (x (t)) - \frac{y (t)}{10}\}$

>	$DEplot3d (DE2, [x (t), y (t)], t = - 10 .. 10, [[x (0) = 1, y (0) = 1]], stepsize = 0.5, title = LC 2, linecolor = magenta)$

$DE2 ≔ \{\frac{&DifferentialD;}{&DifferentialD; t} x (t) = y (t), \frac{&DifferentialD;}{&DifferentialD; t} y (t) = - \sin (x (t)) - \frac{y (t)}{10}\}$

HUE Value

>	$DE3 := \{\frac{&DifferentialD;}{&DifferentialD; x} y (x) = y (x) - z (x), \frac{&DifferentialD;}{&DifferentialD; x} z (x) = z (x) - 2 y (x)\}$

>	$DEplot3d (DE3, [y (x), z (x)], x = 0 .. 3, [[y (0) = 1.638, z (0) = 2.31]], y = 0 .. 2, z = - 4 .. 4, scene = [x, z (x), y (x)], title = LC 3, linecolor = COLOR (HUE, 0.5))$

$DE3 ≔ \{\frac{&DifferentialD;}{&DifferentialD; x} y (x) = y (x) - z (x), \frac{&DifferentialD;}{&DifferentialD; x} z (x) = z (x) - 2 y (x)\}$

RGB Value

>	$DE4 := \{\frac{&DifferentialD;}{&DifferentialD; t} x (t) = y (t), \frac{&DifferentialD;}{&DifferentialD; t} y (t) = - \sin (x (t)) - \frac{y (t)}{10}\}$

>	$DEplot3d (DE4, [x (t), y (t)], t = - 10 .. 10, stepsize = 0.1, [[x (0) = 1, y (0) = 1]], title = LC 4, linecolor = COLOR (RGB, 0.65, 0.89, 0.75))$

$DE4 ≔ \{\frac{&DifferentialD;}{&DifferentialD; t} x (t) = y (t), \frac{&DifferentialD;}{&DifferentialD; t} y (t) = - \sin (x (t)) - \frac{y (t)}{10}\}$

Expression Applied as HUE

>	$DE5 := \{D (x) (t) = y (t), D (y) (t) = - x (t) - y (t)\}$

>	$DEplot3d (DE5, [x (t), y (t)], t = 0 .. 10, [[x (0) = 0, y (0) = 1], [x (0) = 0, y (0) = 0.5]], scene = [t, x (t), y (t)], stepsize = 0.1, title = Damped Oscillations - LC 5, linecolor = [t - \sqrt{t}, \sin (t) - \cos (t)])$

$DE5 ≔ \{D (x) (t) = y (t), D (y) (t) = - x (t) - y (t)\}$

Procedure Applied as HUE

>	$l_p := proc (t) local a, b &semi; a ≔ 10 \cdot \sin (\frac{t}{2}) &semi; b := t &semi; RETURN (b - a) &semi; end &colon;$

>	$DE6 := \{\frac{&DifferentialD;}{&DifferentialD; t} x (t) = - \sin (t), \frac{&DifferentialD;}{&DifferentialD; t} y (t) = \cos (t)\}$

>	$DEplot3d (DE6, [y (t), x (t)], t = - 2 π .. 0, [[y (0) = 0, x (0) = 1]], scene = [t, x (t), y (t)], stepsize = 0.1, title = LC 5, linecolor = l_p)$

$DE6 ≔ \{\frac{&DifferentialD;}{&DifferentialD; t} x (t) = - \sin (t), \frac{&DifferentialD;}{&DifferentialD; t} y (t) = \cos (t)\}$

method

The method option allows the user to make full use of the numerical integration schemes from dsolve/numeric. The default used is classical[rk4]. Consider the following three-dimensional equation plot that uses the classical method.

>	$DE6 := \{D (x) (t) = y (t), D (y) (t) = - x (t) - y (t)\}$

>	$DEplot3d (DE6, [x (t), y (t)], t = 0 .. 10, [[x (0) = 0, y (0) = 1], [x (0) = 0, y (0) = 0.5]], scene = [t, x (t), y (t)], stepsize = 0.1, linecolour = t - \sqrt{t}, method = classical [rk2])$

$DE6 ≔ \{D (x) (t) = y (t), D (y) (t) = - x (t) - y (t)\}$

Now we use the dverk78 method for another plot.

>	$DE7 := \{D (x) (t) = y (t), D (y) (t) = - x (t) - y (t)\}$

>	$DEplot3d (DE7, [x (t), y (t)], t = 0 .. 10, [[x (0) = 0, y (0) = 1], [x (0) = 0, y (0) = 0.5]], scene = [t, x (t), y (t)], stepsize = 0.1, linecolor = t - \sqrt{t}, method = dverk78)$

$DE7 ≔ \{D (x) (t) = y (t), D (y) (t) = - x (t) - y (t)\}$

obsrange

The obsrange option indicates whether the internal integration should stop after one of the user-specified ranges is exceeded. The default is to continue integration along a solution curve, even after the user boundaries have been exceeded. To illustrate this, we graph two DE plots.

>	$DEplot3d (\{\frac{&DifferentialD;}{&DifferentialD; t} x (t) = y (t), \frac{&DifferentialD;}{&DifferentialD; t} y (t) = - \sin (x (t)) - \frac{y (t)}{10}\}, [x (t), y (t)], t = - 10 .. 10, [[x (0) = 1, y (0) = 1]], stepsize = 0.15, x = - 1 .. 1.5, obsrange = false)$

>	$DEplot3d (\{\frac{&DifferentialD;}{&DifferentialD; t} x (t) = y (t), \frac{&DifferentialD;}{&DifferentialD; t} y (t) = - \sin (x (t)) - \frac{y (t)}{10}\}, [x (t), y (t)], t = - 10 .. 10, [[x (0) = 1, y (0) = 1]], stepsize = 0.15, x = - 1 .. 1.5, obsrange = true)$

plot and dsolve/numeric

A variety of plot and dsolve/numeric options are allowed in DEplot3d. Here is an example containing a number of these:

$DEplot3d (\{\frac{&DifferentialD;}{&DifferentialD; t} x (t) = y (t), \frac{&DifferentialD;}{&DifferentialD; t} y (t) = - \sin (x (t))\}, [x (t), y (t)], t = 0 .. 10, [[x (0) = 0, y (0) = 0.5], [x (0) = 0, y (0) = 1], [x (0) = 0, y (0) = 1.8], [x (0) = - 2 π, y (0) = 1], [x (0) = 2 π, y (0) = 0.5], [x (0) = - 2 π, y (0) = 2.1], [x (0) = 2 π, y (0) = - 2.1]], stepsize = 0.2, title = Pendulum Vibrations, orientation = [0, 90], method = classical [abmoulton], corrections = 3, axes = normal, linecolor = \sin (t) - t)$

For more information, consult the DEplot3d help page. Also see the worksheet Plotting Two-Dimensional Differential Equations. You can also refer to the following help pages: diff operator, D operator, classical method, and dverk78 method.

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