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Cell[CellGroupData[{
Cell["An Euler Method Algorithm", "Section"],

Cell[TextData[{
  "The following algorithm implements an Euler Method solution of\n\t",
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  "     with initial condition      ",
  Cell[BoxData[
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  "\nThe stepsize is set to 0.1, and we compute approximate points on the \
solution up until ",
  StyleBox["t",
    FontSlant->"Italic"],
  " = 5.  Execute the cells with the <Enter> key (far right on keyboard; on a \
PC execute with <Ctrl>-<Enter>)."
}], "Text",
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    \(\(\(\ \ \ \)\( (*\ 
      Initialize\ values\ *) \)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\
\(Clear[f, t, y]\[IndentingNewLine]
    \(f[t_, y_] := y\  - \ Sin[t];\)\[IndentingNewLine]
    \(t0\  = \ 0;\)\[IndentingNewLine]
    \(y0\  = \ 0.5;\)\[IndentingNewLine]
    \(h\  = \ 0.1;\)\[IndentingNewLine]
    \(tLast\  = \ 5;\)\[IndentingNewLine]\[IndentingNewLine]
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Cell[BoxData[
    \(\(\( (*\ Start\ separate\ lists\ of\ y - values, \ 
      and\ \((t, 
          y)\)\ points\ through\ which\ the\ solution\ passes . \ \ Both\ \
contain\ just\ our\ initial\ condition\ at\ the\ outset . \ \ Note\ the\ \
curly\ brackets\ throughout, \ rather\ than\ regular\ parentheses, \ 
      which\ are\ never\ used\ for\ anything\ other\ than\ grouping\ in\ \
\(\(Mathematica\)\(.\)\)\ *) \)\(\[IndentingNewLine]\)\(\ \)\(\
\[IndentingNewLine]\)\(\(yVals\  = \ {y0};\)\[IndentingNewLine]
    \(pts\  = \ {{t0, y0}};\)\[IndentingNewLine]\[IndentingNewLine]
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        yCurrent\  + \ 
          h*f[tCurrent, \ yCurrent]; \[IndentingNewLine]tNew\  = \ 
        tCurrent\  + \ h; \[IndentingNewLine]AppendTo[yVals, 
        yNew]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \  (*\ 
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Cell["\<\
The first time you execute the cell above, you will get a number of \
warnings about the names I chose for my variables.  They are not errors, just \
warnings, and if you re-execute the cell, they will go away.\
\>", "Text",
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Cell[TextData[{
  "The previous cell produced two lists, one containing just the approximate \
y-values, and the other containing (",
  StyleBox["t, y",
    FontSlant->"Italic"],
  ") points on the solution curve ",
  StyleBox["y(t)",
    FontSlant->"Italic"],
  ".  One can use the ListPlot[ ] command to show the resulting approximate \
graph of ",
  StyleBox["y(t)",
    FontSlant->"Italic"],
  ".  (ListPlot[ ] is used instead of the Plot[ ] command you may have used \
so often in Math 162.  This is because, rather than having a formula for ",
  StyleBox["y(t)",
    FontSlant->"Italic"],
  ", what we have is a list of distinct points it passes through.)"
}], "Text",
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Cell["One can also use commands like", "Text",
  FontSize->14],

Cell[BoxData[{
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Cell[TextData[{
  "in order to view the very last element in the list yVals (the last y-value \
that was computed) and the 2nd computed point in the list ",
  StyleBox["pts",
    FontSlant->"Italic"],
  " respectively."
}], "Text",
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}, Closed]],

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Cell[TextData[{
  "Finding explicit solutions in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"]
}], "Section"],

Cell[TextData[{
  "The initial-value problem\n\tdy/dt = ",
  Cell[BoxData[
      \(TraditionalForm\`y\  - \ sin(t)\)]],
  ",    y(0) = 0.5\ninvolves a linear ODE and an initial condition.  Being \
among the easier kinds of ODEs to solve explicitly, you would expect that\n   \
   - soon you will be taught how to do so, and\n      - ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " probably has the capability to find the solution.\nIndeed, both are the \
case.  For the purposes of comparisons between the numerical method and the \
actual solution, we will get the actual solution using ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ".  The pertinent command is DSolve[ ].  It may be used in a couple of \
ways.  The cell"
}], "Text",
  FontSize->14],

Cell[BoxData[
    \(DSolve[\(y'\)[t] \[Equal] y[t] - Sin[t], y[t], t]\)], "Input"],

Cell[TextData[{
  "produces a 1-parameter family of solutions.  (The parameter is denoted \
C[1].)  This indicates that there are many solutions of the ODE, one for each \
choice of the parameter.  We have seen this kind of thing before.  As you may \
recall, specifying that we wish for a solution that passes through a specific \
point (in the above case that point is (0, 0.5)) will usually nail down a \
specific solution.  To get ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to find the specific solution that passes through this point, we give it \
not just the DE to solve, but a ",
  StyleBox["list",
    FontSlant->"Italic"],
  " (Note the curly brackets enclosing these equations, and the use of double \
equals signs) of equations to solve, as in the next cell."
}], "Text",
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Cell[BoxData[
    \(DSolve[{\(y'\)[t] \[Equal] y[t] - Sin[t], y[0] \[Equal] 0.5}, y[t], 
      t]\)], "Input"],

Cell[TextData[{
  "You may plot both your approximated points and the actual solution \
together using a command like DisplayTogether[ ], which is not a built-in \
command and must be imported (once per session is all that is required, ",
  StyleBox["prior to the first time you use it",
    FontSlant->"Italic"],
  ") as a part of the \"Graphics\" package."
}], "Text",
  FontSize->14],

Cell[BoxData[{
    \(<< Graphics`Graphics`\), "\[IndentingNewLine]", 
    \(DisplayTogether[ListPlot[pts], 
      Plot[\((Cos[t] + Sin[t])\)/2, {t, 0, 5}]]\)}], "Input"]
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