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Cell[CellGroupData[{
Cell["First-Order Differential Equations and Slope Fields", "Title",
  PageWidth->PaperWidth],

Cell[TextData[StyleBox["Chapter 6, Section 4 (Chapter 5, Section 4 in Early \
Transcendentals)",
  FontFamily->"Arial",
  FontSize->16,
  FontWeight->"Bold"]], "Text",
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Cell[BoxData[{
    \(\(Off[General::spell1]; \)\), 
    \(\[IndentingNewLine]\), 
    \(\(Off[General::spell]; \)\), 
    \(\[IndentingNewLine]\), 
    \(\(slopefield[f_, x_, y_, xmin_, xmax_, ymin_, ymax_, 
        initcond_:  Automatic] := 
      Block[{xloc, yloc, floc, h, k, segs}, 
        \[IndentingNewLine]If[initcond === Automatic, flag = 1, flag = 0]; 
        \[IndentingNewLine]floc[xloc_, yloc_] = 
          f /. {x \[Rule] xloc, y \[Rule] yloc}; 
        \[IndentingNewLine]icloc = initcond /. y \[Rule] u; 
        \[IndentingNewLine]h = \((xmax - xmin)\)/16; 
        \[IndentingNewLine]k = \((ymax - ymin)\)/16; 
        \[IndentingNewLine]segs = 
          Table[Line[
              If[\((Denominator[floc[xloc, yloc]] /. {
                        xloc \[Rule] xmin + i*h, yloc \[Rule] ymin + j*k})\) 
                  \[NotEqual] 
                  0, {{xmin + 
                      \((i - 3/
                              \((10*\@\(1 + floc[xmin + i*h, ymin + j*k]^2\))
                                \))\)*h, 
                    ymin + j*k - 
                      floc[xmin + i*h, ymin + j*k]*3  h/
                          \((10*\@\(1 + floc[xmin + i*h, ymin + j*k]^2\))
                            \)}, {xmin + 
                      \((i + 3/
                              \((10*\@\(1 + floc[xmin + i*h, ymin + j*k]^2\))
                                \))\)*h, 
                    ymin + j*k + 
                      floc[xmin + i*h, ymin + j*k]*3  h/
                          \((10*\@\(1 + floc[xmin + i*h, ymin + j*k]^2\))
                            \)}}, {{xmin + i*h, ymin + \((j - 3/10)\)*k}, {
                    xmin + i*h, ymin + \((j + 3/10)\)*k}}]], {i, 0, 16}, {j, 
              1, 16}]; 
        \[IndentingNewLine]If[flag \[NotEqual] 1, 
          soln = Table[
              \(NDSolve[{\(u'\)[xloc] \[Equal] floc[xloc, u[xloc]], 
                    icloc[\([i]\)]}, u[xloc], {xloc, xmin, xmax}]\)[
                \([1, 1, 2]\)], {i, 1, Length[icloc]}]; 
          \[IndentingNewLine]\(g2 = 
            Plot[Evaluate[soln], {xloc, xmin, xmax}, 
              PlotStyle 
                \[Rule] {{Thickness[0.008], 
                    RGBColor[0.500008, \ 0, \ 0.500008]}}, 
              DisplayFunction \[Rule] Identity]; \)]; 
        \[IndentingNewLine]g1 = 
          Graphics[{RGBColor[0, 0, 1], Thickness[0.005], segs}]; 
        \[IndentingNewLine]If[flag \[NotEqual] 1, 
          \(Show[{g1, g2}, PlotRange \[Rule] {{xmin, xmax}, {ymin, ymax}}, 
            Axes \[Rule] True, AxesLabel \[Rule] {x, y}, 
            ImageSize \[Rule] {72*7, 72*5}, AspectRatio \[Rule] 1, 
            DisplayFunction \[Rule] $DisplayFunction]; \), 
          \(Show[{g1}, PlotRange \[Rule] {{xmin, xmax}, {ymin, ymax}}, 
            Axes \[Rule] True, AxesLabel \[Rule] {x, y}, 
            ImageSize \[Rule] {72*7, 72*5}, AspectRatio \[Rule] 1, 
            DisplayFunction \[Rule] $DisplayFunction]; \)]]; \)\)}], "Input",
  Editable->False,
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Cell[CellGroupData[{

Cell["Introduction", "Section",
  PageWidth->PaperWidth],

Cell["\<\
OBJECTIVE:  Visualize the slope fields and solution curves for selected \
first-order differential equations.\
\>", "Text",
  PageWidth->PaperWidth],

Cell["\<\
This module contains a special command that plots slope fields and selected \
solution curves for first-order differential equations. You can use it to \
obtain solutions for related problems in the text. In addition, you can use \
the slope field command to study a wide variety of first-order differential \
equations and to analyze the long-term behavior of solutions.\
\>", "Text",
  PageWidth->PaperWidth],

Cell[CellGroupData[{

Cell["Technology Guidelines", "Subsection",
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    Background->None],
  StyleBox[" before executing a new module.\nTO OPEN CELLS, put your cursor \
on the right cell bracket and double click.",
    CellFrame->True,
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  "\nINITIALIZATION CELLS\n\tWhen asked if you want to \". . . automatically \
evaluate all the initialization cells in the \tnotebook . . . ,\" respond by \
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Cell["Part I - Drawing Slope Fields and Solution Curves", "Section",
  PageWidth->PaperWidth],

Cell[TextData[{
  "The ",
  StyleBox["slopefield[f_, x_, y_, xmin_, xmax_, ymin_, ymax_, \
initconditions_(optional)]",
    FontWeight->"Bold"],
  " command in the next cell generates a slope field and draws selected \
solution curves for first-order differential equations of the form ",
  Cell[BoxData[
      FormBox[
        StyleBox[\(dy\/dx = f(x, y)\),
          FontSlant->"Italic"], TraditionalForm]]],
  ". The arguments of the command are ",
  StyleBox["f ",
    FontWeight->"Bold"],
  "(the right-hand side function), ",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y)\)]],
  ", ",
  StyleBox["x",
    FontWeight->"Bold"],
  StyleBox[" ",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["(",
    FontWeight->"Bold"],
  "the independent variable), ",
  StyleBox["y",
    FontWeight->"Bold"],
  StyleBox[" ",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["(",
    FontWeight->"Bold"],
  "the unknown function), ",
  StyleBox["xmin",
    FontWeight->"Bold"],
  ", ",
  StyleBox["xmax",
    FontWeight->"Bold"],
  ", ",
  StyleBox["ymin",
    FontWeight->"Bold"],
  ", and ",
  StyleBox["ymax ",
    FontWeight->"Bold"],
  "(the bounds on the slope field), and ",
  StyleBox["initconditions",
    FontWeight->"Bold"],
  " (an optional list of initial conditions). If the list of conditions is \
not included, only the slope field is drawn. Here's how it works for the \
differential equation ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FractionBox[
            StyleBox["dy",
              FontSlant->"Italic"], 
            StyleBox["dx",
              FontSlant->"Italic"]], "=", \(\(\(x\)\(\ \)\)\/y\)}], 
        TraditionalForm]]],
  ". First, we plot only the slope field."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[{
    \(Clear[x, y, f]\), "\n", 
    \(f = x\/y\), "\n", 
    \(\(slopefield[f, x, y, \(-4\), 4, \(-4\), 4];\)\)}], "Input",
  PageWidth->PaperWidth],

Cell["\<\
Now we include a list of initial conditions to obtain some solution curves. \
Each of the four initial conditions will usually give a different solution \
curve.\
\>", "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(slopefield[f, x, y, \(-4\), 4, \(-4\), 
        4, {y[0] \[Equal] 2, y[1] \[Equal] 1, y[1] \[Equal] \(-1\), 
          y[0] \[Equal] \(-2\)}];\)\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "There is a good likelihood that you will get error messages for some \
initial conditions. For example, the solution curve that passes through the \
point ",
  Cell[BoxData[
      \(TraditionalForm\`\((1, 1/2)\)\)]],
  " cannot be expressed as a function when ",
  StyleBox["y ",
    FontSlant->"Italic"],
  "is a function of ",
  StyleBox["x",
    FontSlant->"Italic"],
  ". Here's what happens."
}], "Text",
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Cell[BoxData[
    \(\(slopefield[x/y, x, y, \(-4\), 4, \(-4\), 
        4, {y[1] \[Equal] 1/2}];\)\)], "Input",
  PageWidth->PaperWidth],

Cell["\<\
When this happens, it is probably better to hand sketch some representative \
solution curves on the slope field.\
\>", "Text",
  PageWidth->PaperWidth]
}, Closed]],

Cell[CellGroupData[{

Cell["You Try It: Part I", "Section",
  PageWidth->PaperWidth],

Cell[TextData[{
  "It's fun! Make up some of your own differential equations of the form ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FractionBox[
            StyleBox["dy",
              FontSlant->"Italic"], 
            StyleBox["dx",
              FontSlant->"Italic"]], "=", \(f(x, y)\)}], TraditionalForm]]],
  ", and see what kind of patterns you can generate. Can you form a \
differential equation that will give solution curves that are ellipses? \
hyperbolas? Just try it by changing the function of ",
  StyleBox["x",
    FontSlant->"Italic"],
  " and ",
  StyleBox["y",
    FontSlant->"Italic"],
  " in red."
}], "Text",
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Cell[BoxData[{\(Clear[x, y, f]\), "\n", 
    RowBox[{"f", "=", 
      StyleBox[\(x\/y\),
        FontColor->RGBColor[1, 0, 0]]}], "\n", \(slopefield[f, x, y, \(-4\), 
        4, \(-4\), 4];\)}], "Input",
  PageWidth->PaperWidth]
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Cell[CellGroupData[{

Cell["Part II: Antiderivatives", "Section",
  PageWidth->PaperWidth],

Cell[TextData[{
  "If we consider differential equations of the form ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FractionBox[
            StyleBox["dy",
              FontSlant->"Italic"], 
            StyleBox["dx",
              FontSlant->"Italic"]], "=", \(f(x)\)}], TraditionalForm]]],
  ", then the solutions for ",
  StyleBox["y",
    FontSlant->"Italic"],
  " as a function of ",
  StyleBox["x ",
    FontSlant->"Italic"],
  "are simply the antiderivatives of ",
  Cell[BoxData[
      \(TraditionalForm\`f(x)\)]],
  "; that is, ",
  Cell[BoxData[
      \(TraditionalForm\`y = \[Integral]\(f(x)\) \[DifferentialD]x\)]],
  ". Let's look at some slope fields and solution curves for differential \
equations of this kind."
}], "Text",
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Cell[BoxData[
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        RowBox[{
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                StyleBox["dx",
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Cell[BoxData[{
    \(Clear[f, x, y]\), "\n", 
    \(\(f = x;\)\), "\n", 
    \(\(slopefield[f, x, y, \(-4\), 4, \(-4\), 
        4, {y[0] \[Equal] \(-2\), y[0] \[Equal] 0, 
          y[0] \[Equal] 2}];\)\)}], "Input",
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Cell[TextData[{
  "Notice that the slopes in the field above are constant along any vertical \
line. This shows graphically that the derivative of ",
  StyleBox["y",
    FontSlant->"Italic"],
  " with respect to ",
  StyleBox["x",
    FontSlant->"Italic"],
  " is not a function of ",
  StyleBox["y",
    FontSlant->"Italic"],
  ", as reflected in the differential equation ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FractionBox[
            StyleBox["dy",
              FontSlant->"Italic"], 
            StyleBox["dx",
              FontSlant->"Italic"]], "=", "x"}], TraditionalForm]]],
  ". Along any horizontal line, however, the slopes vary with ",
  StyleBox["x",
    FontSlant->"Italic"],
  ", ",
  "and the pattern shown in the slope field above is consistent with the \
differential equation ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FractionBox[
            StyleBox["dy",
              FontSlant->"Italic"], 
            StyleBox["dx",
              FontSlant->"Italic"]], "=", "x"}], TraditionalForm]]],
  ". The slopes are negative when ",
  StyleBox["x",
    FontSlant->"Italic"],
  " is negative, positive when ",
  StyleBox["x",
    FontSlant->"Italic"],
  " is positive, and zero when ",
  StyleBox["x",
    FontSlant->"Italic"],
  " is zero.\n\nLet's try a periodic function, ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FractionBox[
            StyleBox["dy",
              FontSlant->"Italic"], 
            StyleBox["dx",
              FontSlant->"Italic"]], "=", \(cos\ x\)}], TraditionalForm]]],
  ". "
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[{
    \(\(f = Cos[x];\)\), "\n", 
    \(\(slopefield[f, x, y, \(-Pi\), Pi, \(-Pi\), 
        Pi, {y[0] \[Equal] \(-2\), y[0] \[Equal] 0, 
          y[0] \[Equal] 2}];\)\)}], "Input",
  PageWidth->PaperWidth]
}, Closed]],

Cell[CellGroupData[{

Cell["You Try It: Part II", "Section",
  PageWidth->PaperWidth],

Cell[TextData[{
  "Use the slope field command to study some antiderivative problems of the \
form ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          StyleBox[\(dy\/dx\),
            FontSlant->"Italic"], 
          StyleBox["=",
            FontSlant->"Plain"], 
          StyleBox[\(f(x)\),
            FontSlant->"Italic"]}], TraditionalForm]]],
  " where you pick the function ",
  Cell[BoxData[
      \(TraditionalForm\`f(x)\)]],
  " (in red) and the initial conditions (in red). Don't forget to use the \
double-equals (",
  Cell[BoxData[
      \(TraditionalForm\` == \)]],
  ") if you add initial conditions."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[{\(Clear[x, y, f]\), "\n", 
    RowBox[{
      RowBox[{"f", "=", 
        StyleBox[\(Exp[x]\),
          FontColor->RGBColor[1, 0, 0]]}], ";", "\n", 
      RowBox[{"slopefield", "[", 
        RowBox[{
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          RowBox[{"{", 
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              RowBox[{
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Cell[CellGroupData[{

Cell["Part III: Autonomous Differential Equations", "Section",
  PageWidth->PaperWidth],

Cell[TextData[{
  "Differential equations of the form ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FractionBox[
            StyleBox["dy",
              FontSlant->"Italic"], 
            StyleBox["dx",
              FontSlant->"Italic"]], "=", " ", \(f(y)\)}], 
        TraditionalForm]]],
  " are called autonomous differential equations. Let's look at the \
autonomous differential equation ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          StyleBox[\(dy\/dx\),
            FontSlant->"Italic"], 
          StyleBox["=",
            FontSlant->"Plain"], 
          StyleBox["y",
            FontSlant->"Italic"]}], TraditionalForm]]],
  ". Despite its simplicity, this is surely the most famous of all \
first-order, autonomous differential equations. Let's draw the slope field \
and some solution curves."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[{
    \(Clear[f, x, y]\), "\n", 
    \(\(f = y;\)\), "\n", 
    \(\(slopefield[f, x, y, \(-3\), 3, \(-8\), 
        8, {y[0] \[Equal] \(-2\), y[0] \[Equal] 0, 
          y[0] \[Equal] 2}];\)\)}], "Input",
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Cell[TextData[{
  "Note that in the slope fields for ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FractionBox[
            StyleBox["dy",
              FontSlant->"Italic"], 
            StyleBox["dx",
              FontSlant->"Italic"]], "=", "y"}], TraditionalForm]]],
  ", the slopes are constant along any horizontal line and vary linearly \
along any vertical line. This is the reverse of what we observed for the \
antiderivative problem of the form ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FractionBox[
            StyleBox["dy",
              FontSlant->"Italic"], 
            StyleBox["dx",
              FontSlant->"Italic"]], "=", "x"}], TraditionalForm]]],
  ".\n\nThe logistic equation is another interesting autonomous first-order \
differential equation. Let's draw a slope field and some solution curves for \
a specific one, say ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FractionBox[
            StyleBox["dP",
              FontSlant->"Italic"], 
            StyleBox["dt",
              FontSlant->"Italic"]], "=", \(P(1 - P)\)}], TraditionalForm]]],
  ". The ",
  StyleBox["slopefield",
    FontWeight->"Bold"],
  " command will generate some error messages because two of the solution \
curves go off to infinity and have vertical asymptotes. However, the solution \
curves as drawn are okay."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[{
    \(Clear[f, t, P]\), "\n", 
    \(\(f = P\ \((1 - P)\);\)\), "\n", 
    \(\(slopefield[f, t, P, \(-5\), 5, \(-1\), 
        2, {P[0] \[Equal] \(-0.005\), P[0] \[Equal] 0. , P[0] \[Equal] 0.25, 
          P[0] \[Equal] 0.5, P[0] \[Equal] 0.75, P[0] \[Equal] 1.0, 
          P[0] \[Equal] 1.005}];\)\)}], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "The long-term behavior of the solution to the logistic equation depends \
upon the initial condition. The slope field graph shown above suggests the \
following:  ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(\(lim\)\(\ \)\)\+\(t \[Rule] \[Infinity]\)\) \(P(
            t)\) = \(-\[Infinity]\)\)]],
  " when ",
  Cell[BoxData[
      \(TraditionalForm\`P(0) < 0\)]],
  "; ",
  Cell[BoxData[
      \(TraditionalForm\`P(t) = 0\)]],
  " for all values of ",
  StyleBox["t ",
    FontSlant->"Italic"],
  "when ",
  Cell[BoxData[
      \(TraditionalForm\`P(0) = 0\)]],
  "; and, ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(\ \)\(\(\(\(lim\)\(\ \)\)\+\(t \[Rule] \
\[Infinity]\)\) \(P(t)\) = 1\)\)\)]],
  " when ",
  Cell[BoxData[
      \(TraditionalForm\`P(0) > 0\)]],
  "."
}], "Text",
  PageWidth->PaperWidth]
}, Closed]],

Cell[CellGroupData[{

Cell["You Try It: Part III", "Section",
  PageWidth->PaperWidth],

Cell[TextData[{
  "The logistic equation has another interesting autonomous form. Let's draw \
a slope field and some solution curves for \n\n",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FractionBox[
            StyleBox["dP",
              FontSlant->"Italic"], 
            StyleBox["dt",
              FontSlant->"Italic"]], "=", \(\(-\(P(1 - P)\)\) \((2 - P)\)\)}],
         TraditionalForm]]],
  ". \n\nThe ",
  StyleBox["slopefield",
    FontWeight->"Bold"],
  " command will generate some error messages. See if you can determine why \
this is so after you see the plot. The solution curves are drawn okay."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[{
    \(Clear[f, t, P]\), "\n", 
    \(\(f = \(-P\)\ \((1 - P)\) \((2 - P)\);\)\), "\n", 
    \(\(slopefield[f, t, P, \(-4\), 5, \(-2\), 
        5, {P[0] \[Equal] \(-0.005\), P[0] \[Equal]  .3, 
          P[0] \[Equal] 0.   .7, P[0] \[Equal] 0.5, P[0] \[Equal] 1.3, 
          P[0] \[Equal] 1.7, P[0] \[Equal] 2.3}];\)\)}], "Input",
  PageWidth->PaperWidth],

Cell["\<\
Some solutions approach 2 and some approach 0. What is the dividing line \
between these sets of solutions? What has caused this to happen?

Look up the concepts of a threshold population size and a carrying capacity. \
See how those concepts apply here.\
\>", "Text",
  PageWidth->PaperWidth]
}, Closed]],

Cell[CellGroupData[{

Cell["You Try It: General", "Section",
  PageWidth->PaperWidth],

Cell[TextData[{
  "Consider some nonautonomous differential equations of the form ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FractionBox[
            StyleBox["dy",
              FontSlant->"Italic"], 
            StyleBox["dx",
              FontSlant->"Italic"]], "=", \(f(x, y)\)}], TraditionalForm]]],
  ". In this section we will start with the ",
  StyleBox["slopefield[ ] ",
    FontWeight->"Bold"],
  "command to plot the slope fields, and we will ask you to sketch some \
representative solution curves. Here are a few examples. See if you can \
predict what the solution curves will look like by looking at the direction \
fields.  Describe the long-term behavior of the solutions for all possible \
initial values of ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(y(0)\)\(.\)\)\)]]
}], "Text",
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Cell[CellGroupData[{

Cell[TextData[{
  "Solve  ",
  Cell[BoxData[
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