10.8 Systems of Differential Equations

Mathematica script by Chris Parrish,
   cparrish@sewanee.edu

Sources and references for some of these problems include
    James Stewart, "Calculus: Concepts and Contexts," Second Edition, Brooks/Cole, 2001
    Deborah Hughes-Hallett, Andrew M. Gleason, et. al., "Calculus," Second Edition, John Wiley & Sons, 1998
    Robert Fraga, ed., "Calculus Problems for a New Century," The Mathematical Association of America, 1993

Lancaster's Model for the Battle of Iwo Jima

Hughes-Hallett, Gleason, et al, Exercise 9.8.13, page 550

Here,
  dx/dt = - a y
  dy/dt = - b x
for a,b>0.

In[664]:=

<<Graphics`PlotField` <<Graphics`Colors`

In[666]:=

Clear[f,x,y,t,ax,by,a,b,c,d]

ax = 0.05;
by = 0.01;

f[y_,x_] := 0.2 x/y;

a = 5; b = 60;    (* a <= x <= b *)
c = 5; d = 30;    (* c <= y <= d *)

usForces       = 54;
japaneseForces = 21.5;

pts = {{usForces,japaneseForces}};

field = PlotVectorField[{1,f[y,x]},
                        {x,a,b},{y,c,d},
                        PlotLabel -> "Battle of Iwo Jima\n dy/dx = 0.2 x/y",
                        Axes -> True,
                        AxesLabel -> {"US Forces (1000's)","Japanese Forces (1000's)"},
                        PlotPoints -> 20,
                        Prolog -> ManganeseBlue,
                        Epilog -> {Red,PointSize[0.02],
                                   Map[Point,pts]}];

[Graphics:HTMLFiles/7.6_systemsOfDEs_2.gif]

Lancaster's Square Law for the Battle of Iwo Jima

Hughes-Hallett, Gleason, et al, Exercise 9.9.13, page 550

In[676]:=

Clear[y, x]  RowBox[{RowBox[{y[x_], :=, RowBox[{RowBox[{RowBox[{(, RowBox[{x^2, -, 604 ... Identity] ; <br /> Show[{field, soln}, DisplayFunction$DisplayFunction] ;

[Graphics:HTMLFiles/7.6_systemsOfDEs_4.gif]

Nelson's Battle of Trafalgar (1805)

Hughes-Hallett, Gleason, et al, Exercise 9.9.16, page 551

In[680]:=

Clear[f,x,y,t,ax,by,a,b,c,d]

ax = 1;
by = 1;

f[y_,x_] := x/y;

a = 5; b = 50;    (* a <= x <= b *)
c = 5; d = 50;    (* c <= y <= d *)

britishForces          = 40;
frenchAndSpanishForces = 46;

pts = {{britishForces,frenchAndSpanishForces}};

field = PlotVectorField[{1,f[y,x]},
                        {x,a,b},{y,c,d},
                        PlotLabel -> "Nelson's Battle of Trafalgar\n dy/dx = x/y",
                        Axes -> True,
                        AxesLabel -> {"British (ships)","French and Spanish (ships)"},
                        PlotPoints -> 20,
                        Prolog -> ManganeseBlue,
                        Epilog -> {Red,PointSize[0.02],
                                   Map[Point,pts]},
                        DisplayFunction->Identity];
                                   
Clear[y,x]

y[x_] := Sqrt[x^2 + 516]

soln = Plot[y[x],{x,a,b},
            PlotStyle->Indigo,
            DisplayFunction->Identity];
            
Show[{field,soln},
      DisplayFunction->$DisplayFunction];

[Graphics:HTMLFiles/7.6_systemsOfDEs_5.gif]


Created by Mathematica  (April 25, 2004)