Re: Plotting The Slope field of a differential equation

*To*: mathgroup at smc.vnet.net*Subject*: [mg36484] Re: [mg36430] Plotting The Slope field of a differential equation*From*: Murray Eisenberg <murraye at attbi.com>*Date*: Sun, 8 Sep 2002 03:31:34 -0400 (EDT)*Organization*: Mathematics & Statistics, Univ. of Mass./Amherst*Reply-to*: murray at math.umass.edu*Sender*: owner-wri-mathgroup at wolfram.com

This is a question that's been asked and answered a number of times. One answer, by me, can be found by going to www.wolfram.com and selecting first Resource Library and then MathGroup on the Resource Library page; then search for "direction field". You'll find, among many others, the URL: http://library.wolfram.com/mathgroup/archive/2002/Jul/msg00163.html (This is not to chastise you for not first looking there, but to suggest a way for you to find answers more efficiently than waiting for the post-message/wait-for-response involved in writing to this mailing list.) Here again is what I wrote (for a different ODE, y' = 1/y): ou need to load the package, then manufacture a vector field whose plot is the "slope field" (what you call "vector flow diagram"). To do the latter, let me separately define the function giving the right-hand side of the ODE; you don't really have to do that, but could instead directly use as the first argument to PlotVectorField the right-hand side of my definition of f. << Graphics`PlotField` f[t_, y_] := {1, 1/y} field = PlotVectorField[f[t, y], {t, 0, 3}, {y, 0.05, 3}]; I gave a name to the graphics result just in order to re-display the field below along with the graph of one solution. First, find a general solution: soln = First @ DSolve[{y'[t] == 1/y[t], y[0] == 1}, y[t], t] {y[t] -> Sqrt[2]*Sqrt[1/2 + t]} grf = Plot[y[t] /. soln, {t, 0, 3}]; Now combine the plot with the field: Show[grf, field]; Note that PlotVectorField does not always give quite the result you might want, since the vector lengths are scaled with respect to the magnitude of the actual vectors. You might want just a "direction field" in which all arrows have the same length. You can produce such a graphic by using the ScaleFunction and ScaleFactor options to PlotVectorField. For example: field = PlotVectorField[f[t, y], {t, 0, 3}, {y, 0.05, 3}, ScaleFunction -> (1 &), ScaleFactor -> 0.175]; John M. Resler wrote: Hello once again, ... I have several differential equations I would like to plot in Mathematica. I would like to plot the Slope Fields of them though. Can anyone lead me in the right direction? I can solve the equations trivially but I want to display the slope fields. An example follows : y' + 2y = 3 ... -- Murray Eisenberg murray at math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street Amherst, MA 01375