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66.1 Introduction to plotdf | ||
66.2 Functions and Variables for plotdf |
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The function plotdf
creates a plot of the direction field (also
called slope field) for a first-order Ordinary Differential Equation
(ODE) or a system of two autonomous first-order ODE's.
Plotdf requires Xmaxima. It can be used from the console or any other interface to Maxima, but the resulting file will be sent to Xmaxima for plotting. Please make sure you have installed Xmaxima before trying to use plotdf.
To plot the direction field of a single ODE, the ODE must be written in the form:
dy -- = F(x,y) dx
and the function F should be given as the argument for
plotdf
. If the independent and dependent variables are not x,
and y, as in the equation above, then those two variables should
be named explicitly in a list given as an argument to the plotdf command
(see the examples).
To plot the direction field of a set of two autonomous ODE's, they must be written in the form
dx dy -- = G(x,y) -- = F(x,y) dt dt
and the argument for plotdf
should be a list with the two
functions G and F, in that order; namely, the first
expression in the list will be taken to be the time derivative of the
variable represented on the horizontal axis, and the second expression
will be the time derivative of the variable represented on the vertical
axis. Those two variables do not have to be x and y, but if
they are not, then the second argument given to plotdf must be another
list naming the two variables, first the one on the horizontal axis and
then the one on the vertical axis.
If only one ODE is given, plotdf
will implicitly admit
x=t
, and G(x,y)=1
, transforming the non-autonomous
equation into a system of two autonomous equations.
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[
u,v]
, … options …)
[
dxdt, dydt]
, … options …)
[
dudt, dvdt]
, [
u, v]
, … options …)
Displays a direction field in two dimensions x and y.
dydx, dxdt and dydt are expressions that depend on
x and y. dvdu, dudt and dvdt are
expressions that depend on u and v. In addition to those two
variables, the expressions can also depend on a set of parameters, with
numerical values given with the parameters
option (the option
syntax is given below), or with a range of allowed values specified by a
sliders option.
Several other options can be given within the command, or selected in
the menu. Integral curves can be obtained by clicking on the plot, or
with the option trajectory_at
. The direction of the integration
can be controlled with the direction
option, which can have
values of forward, backward or both. The number of
integration steps is given by nsteps
and the time interval
between them is set up with the tstep
option. The Adams Moulton
method is used for the integration; it is also possible to switch to an
adaptive Runge-Kutta 4th order method.
Plot window menu:
The menu in the plot window has the following options: Zoom, will change the behavior of the mouse so that it will allow you to zoom in on a region of the plot by clicking with the left button. Each click near a point magnifies the plot, keeping the center at the point where you clicked. Holding the Shift key while clicking, zooms out to the previous magnification. To resume computing trajectories when you click on a point, select Integrate from the menu.
The option Config in the menu can be used to change the ODE(s) in use and various other settings. After configuration changes are made, the menu option Replot should be selected, to activate the new settings. If a pair of coordinates are entered in the field Trajectory at in the Config dialog menu, and the enter key is pressed, a new integral curve will be shown, in addition to the ones already shown. When Replot is selected, only the last integral curve entered will be shown.
Holding the right mouse button down while the cursor is moved, can be used to drag the plot sideways or up and down. Additional parameters such as the number of steps, the initial value of t and the x and y centers and radii, may be set in the Config menu.
A copy of the plot can be saved as a postscript file, using the menu option Save.
Plot options:
The plotdf
command may include several commands, each command is
a list of two or more items. The first item is the name of the option,
and the remainder comprises the value or values assigned to the option.
The options which are recognized by plotdf
are the following:
plotdf
, the x
variable will be directly proportional to t.
The default value is 0.1.
tstep
that will be used for the independent variable, to compute an integral
curve.
The default value is 100.
forward
, to make the independent variable increase
nsteps
times, with increments tstep
, backward
, to
make the independent variable decrease, or both
that will lead to
an integral curve that extends nsteps
forward, and nsteps
backward. The keywords right
and left
can be used as
synonyms for forward
and backward
.
The default value is both
.
versus_t
is given any value
different from 0, the second plot window will be displayed. The second
plot window includes another menu, similar to the menu of the main plot
window.
The default value is 0.
name=value
.
name=min:max
Examples:
(%i1) plotdf(exp(-x)+y,[trajectory_at,2,-0.1])$
(%i1) plotdf(x-y^2,[xfun,"sqrt(x);-sqrt(x)"], [trajectory_at,-1,3], [direction,forward], [y,-5,5], [x,-4,16])$
The graph also shows the function y = sqrt(x).
(%i1) plotdf([v,-k*z/m], [z,v], [parameters,"m=2,k=2"], [sliders,"m=1:5"], [trajectory_at,6,0])$
(%i1) plotdf([y,-(k*x + c*y + b*x^3)/m], [parameters,"k=-1,m=1.0,c=0,b=1"], [sliders,"k=-2:2,m=-1:1"],[tstep,0.1])$
(%i1) plotdf([w,-g*sin(a)/l - b*w/m/l], [a,w], [parameters,"g=9.8,l=0.5,m=0.3,b=0.05"], [trajectory_at,1.05,-9],[tstep,0.01], [a,-10,2], [w,-14,14], [direction,forward], [nsteps,300], [sliders,"m=0.1:1"], [versus_t,1])$
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