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Plots the direction field of a first-order ordinary equation (ODE) or
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a system of two autonomous ODE's
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plotdf(expr, ...options...);
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plotdf([expr1,expr2], ...options...);
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In Maxima 5.9.0, plotdf will only work from xmaxima and you cannot use
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the option "trajectory_at". In Maxima 5.9.1, in addition to those two
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limitations there will also be two options missing from the plot menu:
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"Integrate" and "Plot vs t". To solve those problems see
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http://fisica.fe.up.pt/maxima/plotdf/download.html
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In recent CVS versions plotdf should work fine both from xmaxima and maxima.
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plotdf (expr,...,options,..)
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where expr is an expression depending on x and y, which represents the
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right-hand side of the ODE:
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expr can also depend on a set of parameters that must be given numerical
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values with the "parameters" option and those parameters can be changed
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interactively with the "sliders" option (see "PLOTDF OPTIONS" below).
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For numerical values of x, y, and the parameters, float(ev(expr, numer))
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For a system of two autonomous, first-order ODE's, use:
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plotdf ([expr1,expr2],...,options,..)
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Where expr1 and expr2 are two expressions that depend on x and y (but not
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on t), and represent the right-hand side of the ODE's:
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as in the first case, options "parameters" and "sliders" can be used.
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Several options can be given to the plotdf command, or entered into
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the menu that will appear when you click on the upper-left corner of
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the plot window. Integral curves can be obtained by clicking on the
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plot, or with the option "trajectory_at". The direction of the
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integration can be controlled with the "direction" option. The number of
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integration steps is given by "nsteps" and the time interval between
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them is "tstep". The Adams Moulton method is used for the integration;
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it is also possible to switch to an adaptive Runge-Kutta 4th order
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The menu in the plot window has the following options: "Zoom", will
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change the behavior of the mouse so that it will allow you to zoom in
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on a region of the plot by clicking with the left button. Each click
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near a point magnifies the plot, keeping the center at the point where
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you clicked. Holding the SHIFT key while clicking, zooms out to the
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previous magnification. To resume computing trajectories when you
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click on a point, select "Integrate" from the menu.
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To change the ODE(s) in use, or change other settings, select "Config"
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in the menu, enter new values in the dialog window, and then click on
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"Replot" in the main menu bar. If you enter a pair of coordinates in
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the field "Trajectory at" in the Config dialog menu, and press the
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"enter" key, a new integral curve will be shown, in addition to the
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ones already shown. You can change the color before you enter the
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point coordinates. When you select "Replot" all integral curves, except
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the last one, will erased.
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Holding the right mouse button down while you move the cursor allows
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you to drag (translate) the plot sideways or up and down. Additional
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parameters such as the number of steps (nsteps), the initial t value
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(tinitial), and the x and y centers and radii, may be set in the
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You may print to a Postscript printer, or save the plot as a
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postscript file, with the menu option "Save". To switch between
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printing and saving to a Postscript file, select the "Print Options"
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in the dialog window of "Config", change the settings, go back to
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the main menu ("OK" twice) and select "Save".
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Determines the width in x direction of the x values shown by plotdf.
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Defines the height in y direction of the y values shown by plotdf.
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[xcenter,0.0],[ycenter,0.0]
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(xcenter,ycenter) is the origin of the window shown by plotdf.
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Width of plotdf's canvas in pixels.
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Height of plotdf's canvas in pixels.
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[bbox, -2, -1.2, 3.5, 6]
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Bounding box (xmin ymin xmax ymax) of the region shown by plotdf. It
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will override the values o xcenter, ycenter, xradius, yradius.
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The intial value of the t variable used by plotdf to compute integral
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Number of steps to do in one pass by the integrator of plotdf.
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t step size used by plotdf's integrator.
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[xfun,"x^2;sin(x);exp(x)"]
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A semicolon separated list of functions that plotdf will plot on top
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of the direction field.
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The direction (in time) that the integral curves will follow in a
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direction field produced by plotdf. It may be "forward",
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"backward" or "both".
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Any value different from zero will make plotdf open a window showing
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the time dependence of the independent variables in the last integral
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[trajectory_at,0.1,3.2]
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(x,y) coordinates of a point through which an integral curve
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should be shown by plotdf.
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[parameters,"k=1.1,m=2.5"]
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List of parameters, and their numerical values, used in the
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differential equation(s) given to plotdf.
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[sliders,"k=0:4,m=1:3"]
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Ranges of the parameters that plotdf will use to put sliders to
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accept interactive change of the equations parameters.
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To show the direction field of the differential equation
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and the solution that goes through (2, -0.1):
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plotdf(exp(-x)+y,[trajectory_at,2,-0.1]);
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To obtain the direction field for the equation
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and the solution with initial condition y(-1) = 3, we can use the
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plotdf(x-y^2,[xfun,"sqrt(x);-sqrt(x)"],[trajectory_at,-1,3],
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[direction,forward],[yradius,5],[xcenter,6]);
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The following example shows the direction field of a harmonic oscillator,
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and the integral curve through (x,y) = (6,0), with a slider that
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will allow you to change the value of m interactively (k is fixed at
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plotdf([y,-k*x/m],[parameters,"m=2,k=2"],[sliders,"m=1:5"],
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[trajectory_at,6,0]);
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The direction field of a Duffing equation:
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plotdf([y,-(k*x + c*y + l*x^3)/m],[parameters,"k=-1,m=1.0,c=0,l=1"],
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[sliders,"k=-2:2,m=-1:1"],[bbox,-3,-3,3,3],[tstep,0.1]);
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The direction field for a damped pendulum, including the
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solution for the given initial conditions, and with a slider that
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can be used to change the value of the mass m:
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plotdf([y,-g*sin(x)/l - b*y/m/l],[parameters,"g=9.8,l=0.5,m=0.3,b=0.05"],
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[trajectory_at,1.05,-9],[tstep,0.01],[xradius,6],[yradius,14],
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[xcenter,-4],[direction,forward],[nsteps,300],[sliders,"m=0.1:1"]);
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$Id: plotdf.usg,v 1.1 2004/10/30 08:21:08 vvzhy Exp $
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