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Notebook[{

Cell[CellGroupData[{
Cell[TextData[{
  "Differential Equations with ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ":\nNumerical Solutions"
}], "Title",
  TextAlignment->Center,
  TextJustification->0,
  FontSize->18],

Cell[TextData[{
  "In this notebook we'll collect examples of how to use ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to solve differential equations numerically.  Remember:  ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is not a substitute for basic mathematical skills!\n\n",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s function to solve differential equations numerically is NDSolve.  Below \
are some examples.  More detail can be found using the Help Browser under \
NDSolve (including more examples)."
}], "Text"],

Cell[CellGroupData[{

Cell["Clear symbols", "Section",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
In order to avoid interference from symbols defined in other \
notebooks, we first Clear all symbols.  We assume that the relevant symbols \
are in the Global` context.\
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["Clear[\"Global`*\"]", "Input",
  AspectRatioFixed->True]
}, Open  ]],

Cell[CellGroupData[{

Cell["Example 1 -- First-Order ", "Section",
  FontSize->14],

Cell["\<\
First we give a name (de1) to the differential equation.  We use \
the notation y'[x] to indicate dy/dx.  Note that we use = in defining the \
name but == in the actual equation.  We have to specify any constants for a \
numerical solution (unlike using DSolve), which is why converting the \
equation to dimensionless form is often useful.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(de1\  = \ \ \ \(y'\)[x]\  == \ y[x]\  + Cos[2\ x]^2\)], "Input"],

Cell[BoxData[
    RowBox[{
      RowBox[{
        SuperscriptBox["y", "\[Prime]",
          MultilineFunction->None], "[", "x", "]"}], 
      "==", \(Cos[2\ x]\^2 + y[x]\)}]], "Output"]
}, Open  ]],

Cell["\<\
The command to solve the equation numerically is called \"NDSolve\" \
(note the capital letters!).  Here we solve the differential equation with an \
intial condition that y(x=0) = 2, which translates into y[0]==2.  We also \
have to specify the range in x for which a solution is desired; here
we ask for 0 < x < 10.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(solution\  = \ \ NDSolve[\ {de1, \ y[0] == 2}, \ 
        y, \ {x, 0, 10}]\)], "Input"],

Cell[BoxData[
    RowBox[{"{", 
      RowBox[{"{", 
        RowBox[{"y", "\[Rule]", 
          TagBox[\(InterpolatingFunction[{{0.`, 10.`}}, "<>"]\),
            False,
            Editable->False]}], "}"}], "}"}]], "Output"]
}, Open  ]],

Cell[TextData[{
  "The solution for y[x] is in the double set of {}'s in the form of an \
\"interpolating function\".   This is a way of representing the numerical \
solution. \n\nWe can evaluate y[x] at a particular x using ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s substitution function, /.  and using [[1]] to pick out the inner list \
(i.e., to strip away the outer {}'s) :"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(y[0]\  /. \ solution[\([1]\)]\)], "Input"],

Cell[BoxData[
    \(2.`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(y[3]\  /. \ solution[\([1]\)]\)], "Input"],

Cell[BoxData[
    \(50.21695294049395`\)], "Output"]
}, Open  ]],

Cell["\<\
In the first case we checked the intial condition; in the second we \
found y for x=3.\
\>", "Text"],

Cell["It's easiest to define a function y[x] this way:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(y[x_]\  = \ y[x]\  /. \ solution[\([1]\)]\)], "Input"],

Cell[BoxData[
    RowBox[{
      TagBox[\(InterpolatingFunction[{{0.`, 10.`}}, "<>"]\),
        False,
        Editable->False], "[", "x", "]"}]], "Output"]
}, Open  ]],

Cell["\<\
This definition is a bit bizarre because y[x] is on both sides of \
the equation.  But on the right-hand side, y[x] is just a dummy function.  We \
use = instead of := to avoid some problems with delayed evaluation.
Here's how to do it with :=\
\>", "Text"],

Cell[BoxData[
    \(Clear[y]\)], "Input"],

Cell[BoxData[
    \(y[x_]\  := \ Evaluate[y[x]\  /. \ solution[\([1]\)]\ ]\)], "Input"],

Cell["Now we can plot:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell["We can also find when y=10 using FindRoot:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FindRoot[y[x] == 10\ , {x, 1}]\)], "Input"],

Cell[BoxData[
    \({x \[Rule] 1.431649495647738`}\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Example 2 -- Second-Order ", "Section",
  FontSize->14],

Cell["\<\
Now we consider a second-order equation.  The procedure is the \
same, except now we have
to specify two initial conditions.  We'll use the equation for a ball tossed \
up with air resistance included.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(de2\  = \ \ \ \(\(z'\)'\)[
          t]\  == \ \(-g\)\  - \(\(b\)\(\ \)\(\(z'\)[t]\)\(\ \)\)\)], "Input"],

Cell[BoxData[
    RowBox[{
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      RowBox[{\(-g\), "-", 
        RowBox[{"b", " ", 
          RowBox[{
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              MultilineFunction->None], "[", "t", "]"}]}]}]}]], "Output"]
}, Open  ]],

Cell["We have to specify g and b and initial z and v:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(g\  = \ 9.8\)], "Input"],

Cell[BoxData[
    \(9.8`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(z0\  = \ 0\)], "Input"],

Cell[BoxData[
    \(0\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(v0\  = \ 20\)], "Input"],

Cell[BoxData[
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}, Open  ]],

Cell["Now solve with those initial conditions specified:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(solution2\  = \ \ NDSolve[\ {de2, z[0] == z0, \ \(z'\)[0] == v0}, 
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Cell[BoxData[
    RowBox[{"{", 
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Cell["\<\
The solution for y[t] is in the double set of {}'s in the form of \
an \"interpolating function\".   This is a way of representing the numerical \
solution. 

Evaluate z[t] at t=0 to check initial conditions:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell["\<\

We can plot the result if we use the Evaluate function:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Plot[
      Evaluate[\ z[t]\  /. \ solution2[\([1]\)]], \ {t, 0, 5}]\)], "Input"],

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