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Notebook[{

Cell[CellGroupData[{
Cell["Regular Power Series Solution", "Title"],

Cell["\<\
Solution of
y''+cos(x) y=0
for y(x) with y(0)=1 and y'(0)=0.
An exact solution can be found in this case, and can
be compared with the series solution.\
\>", "Text"],

Cell[BoxData[
    \(Off[General::spell1]\)], "Input"],

Cell[TextData[{
  "Let's find a solution valid out to ",
  Cell[BoxData[
      \(TraditionalForm\`x\^10\)]],
  ".  The first step in finding a series solution is to find an expansion for \
Cos[x]: "
}], "Text"],

Cell[BoxData[
    \(\(n = 10;\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(cosser = Cos[x] + O[x]^\((n + 1)\)\)], "Input"],

Cell[BoxData[
    InterpretationBox[
      RowBox[{
      "1", "-", \(x\^2\/2\), "+", \(x\^4\/24\), "-", \(x\^6\/720\), 
        "+", \(x\^8\/40320\), "-", \(x\^10\/3628800\), "+", 
        InterpretationBox[\(O[x]\^11\),
          SeriesData[ x, 0, {}, 0, 11, 1]]}],
      SeriesData[ x, 0, {1, 0, 
        Rational[ -1, 2], 0, 
        Rational[ 1, 24], 0, 
        Rational[ -1, 720], 0, 
        Rational[ 1, 40320], 0, 
        Rational[ -1, 3628800]}, 0, 11, 1]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(y[x_] = Sum[a[k]\ x^k, \ {k, 0, n, 2}] + O[x]^\((n + 1)\)\)], "Input"],

Cell[BoxData[
    InterpretationBox[
      RowBox[{\(a[0]\), "+", \(a[2]\ x\^2\), "+", \(a[4]\ x\^4\), 
        "+", \(a[6]\ x\^6\), "+", \(a[8]\ x\^8\), "+", \(a[10]\ x\^10\), "+", 
        
        InterpretationBox[\(O[x]\^11\),
          SeriesData[ x, 0, {}, 0, 11, 1]]}],
      SeriesData[ x, 0, {
        a[ 0], 0, 
        a[ 2], 0, 
        a[ 4], 0, 
        a[ 6], 0, 
        a[ 8], 0, 
        a[ 10]}, 0, 11, 1]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(deqn = \(y''\)[x] + cosser*y[x] \[Equal] 0\)], "Input"],

Cell[BoxData[
    RowBox[{
      InterpretationBox[
        RowBox[{\((a[0] + 2\ a[2])\), 
          "+", \(\((\(-\(a[0]\/2\)\) + a[2] + 12\ a[4])\)\ x\^2\), 
          "+", \(\((a[0]\/24 - a[2]\/2 + a[4] + 30\ a[6])\)\ x\^4\), 
          "+", \(\((\(-\(a[0]\/720\)\) + a[2]\/24 - a[4]\/2 + a[6] + 
                56\ a[8])\)\ x\^6\), 
          "+", \(\((a[0]\/40320 - a[2]\/720 + a[4]\/24 - a[6]\/2 + a[8] + 
                90\ a[10])\)\ x\^8\), "+", 
          InterpretationBox[\(O[x]\^9\),
            SeriesData[ x, 0, {}, 0, 9, 1]]}],
        SeriesData[ x, 0, {
          Plus[ 
            a[ 0], 
            Times[ 2, 
              a[ 2]]], 0, 
          Plus[ 
            Times[ 
              Rational[ -1, 2], 
              a[ 0]], 
            a[ 2], 
            Times[ 12, 
              a[ 4]]], 0, 
          Plus[ 
            Times[ 
              Rational[ 1, 24], 
              a[ 0]], 
            Times[ 
              Rational[ -1, 2], 
              a[ 2]], 
            a[ 4], 
            Times[ 30, 
              a[ 6]]], 0, 
          Plus[ 
            Times[ 
              Rational[ -1, 720], 
              a[ 0]], 
            Times[ 
              Rational[ 1, 24], 
              a[ 2]], 
            Times[ 
              Rational[ -1, 2], 
              a[ 4]], 
            a[ 6], 
            Times[ 56, 
              a[ 8]]], 0, 
          Plus[ 
            Times[ 
              Rational[ 1, 40320], 
              a[ 0]], 
            Times[ 
              Rational[ -1, 720], 
              a[ 2]], 
            Times[ 
              Rational[ 1, 24], 
              a[ 4]], 
            Times[ 
              Rational[ -1, 2], 
              a[ 6]], 
            a[ 8], 
            Times[ 90, 
              a[ 10]]]}, 0, 9, 1]], "==", "0"}]], "Output"]
}, Open  ]],

Cell[TextData[{
  "Here are the unknowns in the expansion of ",
  Cell[BoxData[
      \(TraditionalForm\`y(x)\)]],
  " that we must solve for:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(coeffs = Table[a[k], {k, 2, n, 2}]\)], "Input"],

Cell[BoxData[
    \({a[2], a[4], a[6], a[8], a[10]}\)], "Output"]
}, Open  ]],

Cell["Here are the algebraic equations for those coefficients:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(eqns = LogicalExpand[deqn]\)], "Input"],

Cell[BoxData[
    \(a[0] + 2\ a[2] == 0 && \(-\(a[0]\/2\)\) + a[2] + 12\ a[4] == 0 && 
      a[0]\/24 - a[2]\/2 + a[4] + 30\ a[6] == 
        0 && \(-\(a[0]\/720\)\) + a[2]\/24 - a[4]\/2 + a[6] + 56\ a[8] == 0 && 
      a[0]\/40320 - a[2]\/720 + a[4]\/24 - a[6]\/2 + a[8] + 90\ a[10] == 
        0\)], "Output"]
}, Open  ]],

Cell["\<\
We actually don't need the TableForm for the math, it is just nice \
to look at:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TableForm[ReplaceAll[eqns, And \[Rule] List]]\)], "Input"],

Cell[BoxData[
    InterpretationBox[GridBox[{
          {\(a[0] + 2\ a[2] == 0\)},
          {\(\(-\(a[0]\/2\)\) + a[2] + 12\ a[4] == 0\)},
          {\(a[0]\/24 - a[2]\/2 + a[4] + 30\ a[6] == 0\)},
          {\(\(-\(a[0]\/720\)\) + a[2]\/24 - a[4]\/2 + a[6] + 56\ a[8] == 
              0\)},
          {\(a[0]\/40320 - a[2]\/720 + a[4]\/24 - a[6]\/2 + a[8] + 90\ a[10] == 
              0\)}
          },
        RowSpacings->1,
        ColumnSpacings->3,
        RowAlignments->Baseline,
        ColumnAlignments->{Left}],
      TableForm[ {
        Equal[ 
          Plus[ 
            a[ 0], 
            Times[ 2, 
              a[ 2]]], 0], 
        Equal[ 
          Plus[ 
            Times[ 
              Rational[ -1, 2], 
              a[ 0]], 
            a[ 2], 
            Times[ 12, 
              a[ 4]]], 0], 
        Equal[ 
          Plus[ 
            Times[ 
              Rational[ 1, 24], 
              a[ 0]], 
            Times[ 
              Rational[ -1, 2], 
              a[ 2]], 
            a[ 4], 
            Times[ 30, 
              a[ 6]]], 0], 
        Equal[ 
          Plus[ 
            Times[ 
              Rational[ -1, 720], 
              a[ 0]], 
            Times[ 
              Rational[ 1, 24], 
              a[ 2]], 
            Times[ 
              Rational[ -1, 2], 
              a[ 4]], 
            a[ 6], 
            Times[ 56, 
              a[ 8]]], 0], 
        Equal[ 
          Plus[ 
            Times[ 
              Rational[ 1, 40320], 
              a[ 0]], 
            Times[ 
              Rational[ -1, 720], 
              a[ 2]], 
            Times[ 
              Rational[ 1, 24], 
              a[ 4]], 
            Times[ 
              Rational[ -1, 2], 
              a[ 6]], 
            a[ 8], 
            Times[ 90, 
              a[ 10]]], 0]}]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(solset = Solve[eqns, coeffs]\)], "Input"],

Cell[BoxData[
    \({{a[2] \[Rule] \(-\(a[0]\/2\)\), a[4] \[Rule] a[0]\/12, 
        a[6] \[Rule] \(-\(a[0]\/80\)\), a[8] \[Rule] \(11\ a[0]\)\/8064, 
        a[10] \[Rule] \(-\(\(17\ a[0]\)\/129600\)\)}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

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

Cell[BoxData[
    InterpretationBox[
      RowBox[{\(a[0]\), "-", \(1\/2\ a[0]\ x\^2\), "+", \(1\/12\ a[0]\ x\^4\),
         "-", \(1\/80\ a[0]\ x\^6\), "+", \(\(11\ a[0]\ x\^8\)\/8064\), 
        "-", \(\(17\ a[0]\ x\^10\)\/129600\), "+", 
        InterpretationBox[\(O[x]\^11\),
          SeriesData[ x, 0, {}, 0, 11, 1]]}],
      SeriesData[ x, 0, {
        a[ 0], 0, 
        Times[ 
          Rational[ -1, 2], 
          a[ 0]], 0, 
        Times[ 
          Rational[ 1, 12], 
          a[ 0]], 0, 
        Times[ 
          Rational[ -1, 80], 
          a[ 0]], 0, 
        Times[ 
          Rational[ 11, 8064], 
          a[ 0]], 0, 
        Times[ 
          Rational[ -17, 129600], 
          a[ 0]]}, 0, 11, 1]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(yss[x_] = Normal[ys[x]]\)], "Input"],

Cell[BoxData[
    \(a[0] - 1\/2\ x\^2\ a[0] + 1\/12\ x\^4\ a[0] - 
      1\/80\ x\^6\ a[
          0] + \(11\ x\^8\ a[0]\)\/8064 - \(17\ x\^10\ a[0]\)\/129600\)], \
"Output"]
}, Open  ]],

Cell["\<\
So here is the series solution that satisfies the specified initial \
conditions:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(yss1[x_] = yss[x] /. a[0] \[Rule] 1\)], "Input"],

Cell[BoxData[
    \(1 - x\^2\/2 + x\^4\/12 - 
      x\^6\/80 + \(11\ x\^8\)\/8064 - \(17\ x\^10\)\/129600\)], "Output"]
}, Open  ]],

Cell["Now find exact solution:", "Text"],

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

Cell[CellGroupData[{

Cell[BoxData[
    \(exact = 
      DSolve[{\(y''\)[x] + Cos[x]\ y[x] \[Equal] 0, \(y'\)[0] \[Equal] 0, 
          y[0] == 1}, y[x], x]\)], "Input"],

Cell[BoxData[
    \({{y[x] \[Rule] 
          MathieuC[0, \(-2\), x\/2]\/MathieuC[0, \(-2\), 0]}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

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

Cell[BoxData[
    \(MathieuC[0, \(-2\), x\/2]\/MathieuC[0, \(-2\), 0]\)], "Output"]
}, Open  ]],

Cell["Plot approximate (red) and exact (green) solutions:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Plot[{ey[x], yss1[x]}, {x, 0, 10}, PlotRange \[Rule] {\(-10\), 10}, 
      PlotStyle \[Rule] {RGBColor[0, 1, 0], \ RGBColor[1, 0, 0]}]\)], "Input"],

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