(*^

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                                    Visualizing Differential Equations

The cell below defines a Mathematica   procedure  SlopeField  that draws slope fields for a differential equation of the form

                                                             y' = f(x, y)

Evaluate it now.
;[s]
10:0,1;70,0;97,2;108,0;122,1;132,0;260,1;272,0;274,3;289,0;291,-1;
4:5,13,9,Times,0,12,0,0,0;3,13,9,Times,1,12,0,0,0;1,13,9,Times,2,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
SlopeField[foo_, xlow_, xhigh_, ylow_, yhigh_] :=
 Block[{slopes = {}, 
        hx = (xhigh - xlow)/20.0,
        hy = (yhigh - ylow)/20.0,
        dx = (xhigh - xlow)/8.0,
        dy = (yhigh - ylow)/8.0,
        x, y, i, j, m
       }, 
   For[x = xlow; i = 0, 
       i <= 8, 
       ++i; x = x + dx,
       For[y = ylow; j = 0,
           j <= 8, 
           ++j; y = y + dy,
           m = foo[x, y];
           If[Abs[m * hx] <= hy, kx = hx, 
                                 kx = hy/Abs[m]];
           slopes = Join[{Point[{x, y}]}, slopes];
           If[NumberQ[m], 
              slopes = Join[{Line[{{x-kx, y-m*kx}, 
                                   {x+kx, y+m*kx}}]}, 
                             slopes]]
          ]
      ];
   Show[Graphics[{PointSize[0.01], slopes}],
        Axes -> Automatic,
        AspectRatio -> 1.0,
        DisplayFunction -> $DisplayFunction]
 ]

:[font = special1; inactive; dontPreserveAspect; ]
The following cell shows how  SlopeField  can graph a slope field like the ones in the browser window.  Evaluate it now.  When you evaluate this cell it will generate warning messages because the right hand side of this differential equation is undefined when x = 0.
;[s]
9:0,0;29,1;30,2;41,1;42,0;104,3;119,0;122,3;265,0;267,-1;
4:4,13,9,Times,0,12,0,0,0;2,13,9,Times,2,12,0,0,0;1,13,9,Times,1,12,0,0,0;2,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
Clear[fcn]
fcn[x_, y_] := y/x
SlopeField[fcn, -3, 3, -3, 3] 
:[font = special1; inactive; dontPreserveAspect; ]
Notice that the function  f[x, y]  on the right hand side of the differential equation

                       dy/dx = f[x, y]

must be specified in a line of its own rather than as part of the  SlopeField  line.  In other words, 

           WRONG!    SlopeField[-x/y, -1.0, 1.0, -1.0, 1.0]     WRONG!

will not work.  In addition, the first variable must be the independent variable and the second variable must be the dependent variable.  For example, the differential equation

                                              dp/dt = p(1 - p)

should be entered as

                 fcn[t_, p_] := p(1 - p)    NOT    fcn[p_, t_] := p(1 - p)

The form of the  SlopeField  statement is

SlopeField[function, xlow, xhigh, ylow, yhigh]

where  xlow  and  xhigh  give the range for the variable x, and  ylow  and  yhigh  give the range for the variable y.
;[s]
36:0,0;26,1;33,0;111,1;126,0;195,1;205,0;243,3;302,0;364,1;375,0;421,1;430,0;528,1;544,0;585,1;608,0;612,1;642,2;643,0;661,1;671,0;687,1;733,0;742,1;746,0;753,1;758,0;792,1;793,0;800,1;804,0;811,1;816,0;850,1;851,0;853,-1;
4:18,13,9,Times,0,12,0,0,0;16,13,9,Times,1,12,0,0,0;1,13,9,Times,0,12,65535,0,0;1,13,9,Times,1,12,65535,0,65535;
^*)