66.1 Introduction to plotdf

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.

66.2 Functions and Variables for plotdf

Function: plotdf (dydx, … options …) ¶

Function: plotdf (dvdu, [u,v], … options …) ¶

Function: plotdf ([dxdt, dydt], … options …) ¶

Function: plotdf ([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:

• tstep defines the length of the increments on the independent variable t, used to compute an integral curve. If only one expression dydx is given to plotdf, the x variable will be directly proportional to t. The default value is 0.1.
• nsteps defines the number of steps of length tstep that will be used for the independent variable, to compute an integral curve. The default value is 100.
• direction defines the direction of the independent variable that will be followed to compute an integral curve. Possible values are 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.
• tinitial defines the initial value of variable t used to compute integral curves. Since the differential equations are autonomous, that setting will only appear in the plot of the curves as functions of t. The default value is 0.
• versus_t is used to create a second plot window, with a plot of an integral curve, as two functions x, y, of the independent variable t. If 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.
• trajectory_at defines the coordinates xinitial and yinitial for the starting point of an integral curve. The option is empty by default.
• parameters defines a list of parameters, and their numerical values, used in the definition of the differential equations. The name and values of the parameters must be given in a string with a comma-separated sequence of pairs name=value.
• sliders defines a list of parameters that will be changed interactively using slider buttons, and the range of variation of those parameters. The names and ranges of the parameters must be given in a string with a comma-separated sequence of elements name=min:max
• xfun defines a string with semi-colon-separated sequence of functions of x to be displayed, on top of the direction field. Those functions will be parsed by Tcl and not by Maxima.
• x should be followed by two numbers, which will set up the minimum and maximum values shown on the horizontal axis. If the variable on the horizontal axis is not x, then this option should have the name of the variable on the horizontal axis. The default horizontal range is from -10 to 10.
• y should be followed by two numbers, which will set up the minimum and maximum values shown on the vertical axis. If the variable on the vertical axis is not y, then this option should have the name of the variable on the vertical axis. The default vertical range is from -10 to 10.

Examples:

• To show the direction field of the differential equation y' = exp(-x) + y and the solution that goes through (2, -0.1):

  (%i1) plotdf(exp(-x)+y,[trajectory_at,2,-0.1])$
  

  
• To obtain the direction field for the equation diff(y,x) = x - y^2 and the solution with initial condition y(-1) = 3, we can use the command:

  (%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).

  
• The following example shows the direction field of a harmonic oscillator, defined by the two equations dz/dt = v and dv/dt = -k*z/m, and the integral curve through (z,v) = (6,0), with a slider that will allow you to change the value of m interactively (k is fixed at 2):

  (%i1) plotdf([v,-k*z/m], [z,v], [parameters,"m=2,k=2"],
             [sliders,"m=1:5"], [trajectory_at,6,0])$
  

  
• To plot the direction field of the Duffing equation, m*x''+c*x'+k*x+b*x^3 = 0, we introduce the variable y=x' and use:

  (%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])$
  

  
• The direction field for a damped pendulum, including the solution for the given initial conditions, with a slider that can be used to change the value of the mass m, and with a plot of the two state variables as a function of time:

  (%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])$