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Notebook[{
Cell["\<\
Solution of ODE's and eigenvalue problems with a Chebyshev \
polynomial spectral method\
\>", "Title"],

Cell[TextData[{
  "This notebook has been written in ",
  StyleBox["Mathematica ",
    FontSlant->"Italic"],
  "by \n\n",
  StyleBox[
  "Mark J. McCready\nProfessor and Chair of Chemical Engineering\nUniversity \
of Notre Dame\nNotre Dame IN 46556\nUSA",
    FontSize->14],
  "\n\nMark.J.McCready.1@nd.edu\n",
  ButtonBox["http://www.nd.edu/~mjm/",
    ButtonData:>{
      URL[ "http://www.nd.edu/~mjm/"], None},
    ButtonStyle->"Hyperlink"],
  "\n\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \
the World Wide Web and the Internet.\n\nI would hope that as a professional \
courtesy, this notice remain visible to other users. \nThere is no charge for \
copying and dissemination   \n\nVersion:  5/17/99\nMore recent versions of \
this notebook should be available at the web site:\n",
  ButtonBox["http://www.nd.edu/~mjm/spectral.ode.eigens.nb",
    ButtonData:>{
      URL[ "http://www.nd.edu/~mjm/spectral.ode.eigens.nb"], None},
    ButtonStyle->"Hyperlink"]
}], "Text"],

Cell[CellGroupData[{

Cell["Summary", "Subtitle"],

Cell[TextData[{
  "This notebook demonstrates the Orszag-tau ( a modification of the \
Galerkin) spectral method for a simple ODE and an ODE eigenvalue problem.  It \
is intended as a first introduction to solving these problems with a spectral \
numerical method.\n\n",
  StyleBox["Reference:",
    FontSize->10,
    FontWeight->"Bold"],
  StyleBox[
  "  S. A. Orszag (1971) \"Accurate solution of the Orr-Sommerfeld stability \
equation\", Journal of Fluid Mechanics, ",
    FontSize->10],
  StyleBox["50",
    FontSize->10,
    FontWeight->"Bold"],
  StyleBox[" pp 689-703.\n",
    FontSize->10],
  "\nThe coefficients of the algebraic equations are computed directly from \
the orthogonality properties of the Chebyshev polynomials using \
orthogonality.\n\n",
  StyleBox["Reference:",
    FontSize->10,
    FontWeight->"Bold"],
  StyleBox[
  "  R. Miesen and B. J. Boersma (1995) \"Hydrodynamic stability of a sheared \
liquid film\", Journal of Fluid Mechanics, ",
    FontSize->10],
  StyleBox["301",
    FontSize->10,
    FontWeight->"Bold"],
  StyleBox[" pp 175-202.",
    FontSize->10]
}], "Text"],

Cell[CellGroupData[{

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " aside"
}], "Subsection"],

Cell[TextData[{
  "In ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ", it is convenient to give all expressions a \"name\".  I try to pick ones \
that are consistent with what is being done (but sometimes \"temp\" is used). \
 This assignment is done with an \"=\" sign.  To make an equation, a \"==\" \
is used.  This distinction is very useful in computer algebra and is employed \
in all of the packages with which I am familiar ."
}], "Text"],

Cell[CellGroupData[{

Cell["Input notation", "Subsubsection"],

Cell["\<\
I have used a mix of input notation to show how it can be done. \
\
\>", "Text"],

Cell["\<\
I would enter the dynamic boundary condition from a key pad as\
\>",
   "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dc = 
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              g\ \[Eta][x, t])\) + \n\t\t\t
        \(\[Rho]\_2\) 
          \((\(-D[\[Phi]\_2[x, y, t], t]\) - \n\t\t\t\t\t
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Cell[TextData[{
  "In ",
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this practical.  "
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Cell[TextData[{
  "Finally it may be more easily ",
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  " as"
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Cell["\<\
Note that you can convert any \"cryptic\" input expression by \
selecting the cell, going up to the cell menu and selecting \"convert to \
traditional form\".  This is \"shift curly t\" on the keypad.  \
\>", "Text"]
}, Closed]]
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Cell[CellGroupData[{

Cell["Some properties of Chebyshev polynomials", "Subtitle"],

Cell[CellGroupData[{

Cell["\<\
For this example, we will be using the ChebyshevT polynomials which \
are orthogonal and 
non singular on the interval [-1,1].  They also have endpoint values @y=1 of \
1, and @ y=-1, of 1 or -1 depending if they have even or odd index numbers.
  
Let's see what these are\
\>", "Text",
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      64\ y\^7 - 112\ y\^5 + 56\ y\^3 - 7\ y, 
      128\ y\^8 - 256\ y\^6 + 160\ y\^4 - 32\ y\^2 + 1}\)], "Output"]
}, Open  ]],

Cell[TextData[
"If we wish, we can plot all of these.  \nThe Evaluate is needed to make the \
function be recognized before it is numerically evaluated."], "Text",
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Cell[CellGroupData[{

Cell[BoxData[
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This is too complicated to see much.  But there are even \
polynomials and odd ones with the expected symmetry.
Also, it can be seen that the interval is spanned effectively with these so \
they look to be good expansion functions.\
\>", "Text",
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2/\[Pi]i/c[i].\n  The c[i]=1 for all i except i=0 when it equals 2."], "Text",\

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Cell["\<\
Basic idea of how to use a spectral numerical method to solve an \
ODE.\
\>", "Subtitle"],

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Cell[TextData[{
  "You probably know from that it is possible to fit an arbitrary function \
using a set of orthogonal polynomials by forming the inner product to \
evaluate the coefficients.  (If you need a basic introduction to spectral \
theory you my want to check out ",
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  " .)  We thus ",
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  " that we could solve an ode, which must have some function form (albeit) \
unknown, with an appropriate set of orthogonal polynomials, or other \
orthogonal functions.  The obvious questions are which functions and how do \
we get the cofficients.\n\nThe starting point is an assumed solution form of:\
\n\n",
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  "  =",
  Cell[BoxData[
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  " to be orthogonal on the domain of interest and capable of producing a \
boundary conditions of the needed form.  \n\nWe will substitute the assumed \
form for the solution into the ODE and boundary conditions, form an inner \
product with every term (not the boundary conditions however) (why not), and \
then see if we can evaluate the coefficients.  If we can, we have a solution.\
\n\nNote however, that we will expect to use a finite number of terms -- as \
opposed to the ",
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all terms.  We then hope that the solution will get more accurate as we \
increase the number of terms.\n\nWe first start with a very simple ode that \
has an analytical solution with which to compare.      "
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Cell["Solution of a simple ode", "Subtitle"],

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Cell[TextData["Analytical solution"], "Subsection",
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Cell["\<\
Here is a simple differential equation that we can get an analytic \
solution to. 
 It will be useful for comparison when we solve it with a spectral method.
 
 Note it is first written in \"traditional form\" and the cell is inactive.\
\
\>", "Text",
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Cell["Now we solve the ODE.", "Text"],

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Cell[CellGroupData[{

Cell[TextData["Numerical solution"], "Subsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
Now solve the equation with a Galerkin spectral technique that \
involves substituting a series of Chebyshevs into the differential equation, \
employing orthogonality and then solving the algebraic equations. We use \
direct substitution and don't attempt to reduce the derivatives with \
analytical expressions.
 
 Note it is first written in \"traditional form\" and the cell is inactive.\
\
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    FormBox[
      RowBox[{"ode1", "=", 
        RowBox[{
          FractionBox[\(\[PartialD]\^2\( vv(y)\)\), \(\[PartialD]y\^2\),
            MultilineFunction->None], "+", \(vv(y)\)}]}], TraditionalForm]], 
  "Input",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["Here we start the numerical solution.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(ode1 = \[PartialD]\_{y, 2}vv[y] + vv[y]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    FormBox[
      RowBox[{\(vv(y)\), "+", 
        RowBox[{
          SuperscriptBox["vv", "\[DoublePrime]",
            MultilineFunction->None], "(", "y", ")"}]}], TraditionalForm]], 
  "Output"]
}, Open  ]],

Cell[TextData[{
  "The basic assumption of a Galerkin spectral method is that the numerical \
solution, can be written as a sum of orthogonal functions.  This trial \
solution has a finite number of terms and is valid over the entire domain.  \
The functions are chosen to automatically fit the boundary conditions (e.g., \
perhaps by some clever choices of sums of groups of polynomials, or shear \
luck).\n\nIf the functions do not assure automatic fitting of the boundary \
conditions, we will have to make some adjustments.  The \"tau\" method, used \
below, replaces one of the equations for a[i] generated by the ODE, for ",
  StyleBox["each",
    FontSlant->"Italic"],
  " algebraic equation that is needed to assure boundary condition fitting.  \
",
  " Stated again, if we have 4 boundary conditions, we will replace 4 of the \
equations generated from the ode with the 4 equations that must be fit on the \
boundary.",
  "  To minimize the error, we think of this as starting with more terms in \
the sum than we need, then dropping of the highest \"frequency\" modes that \
are assumed (this is confirmed) to be less important to getting an accurate \
solution than the lowest modes.\n\nHere is where we substitute Chebyshev \
polynomials for vv[y]."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp1 = 
      ode1 /. {vv[y] \[Rule] a[i]\ ChebyshevT[i, y], 
          \[PartialD]\_{y, a1_}vv[y] \[Rule] 
            \[PartialD]\_{y, a1}\((a[i]\ ChebyshevT[i, y])\)}\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\(a(i)\)\ \(\(T\_i\)(y)\) + 
      \(i\ \(a(i)\)\ 
          \((i\ \(\(T\_i\)(y)\) - y\ \(\(U\_\(i - 1\)\)(y)\))\)\)\/\(y\^2 - 1
          \)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp2 = ExpandAll[temp1]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\(\(a(i)\)\ \(\(T\_i\)(y)\)\ i\^2\)\/\(y\^2 - 1\) - 
      \(y\ \(a(i)\)\ \(\(U\_\(i - 1\)\)(y)\)\ i\)\/\(y\^2 - 1\) + 
      \(a(i)\)\ \(\(T\_i\)(y)\)\)], "Output"]
}, Open  ]],

Cell["Now get a finite size sum to solve.   ", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(\(n = 12; \)\)], "Input",
  AspectRatioFixed->True],

Cell["\<\
Because of the symmetry of the boundary conditions and the \
occurence of only even derivatives, the solution will be an even function so \
we need only even polynomials.

This sum, with appropriate values for the a(i)'s would have a value of \
approximately 0 and thus solve the ODE. \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{"temp3", "=", 
      RowBox[{
        UnderoverscriptBox["\[Sum]", GridBox[{
              {\(i = 0\)},
              {\(\[CapitalDelta]\[MediumSpace]i = 2\)}
              }], "n"], "temp2"}]}]], "Input",
  AspectRatioFixed->True,
  CellTags->"numsum"],

Cell[BoxData[
    \(TraditionalForm
    \`\(-\(\(4\ \(a(2)\)\ y\^2\)\/\(y\^2 - 1\)\)\) - 
      \(4\ \((8\ y\^3 - 4\ y)\)\ \(a(4)\)\ y\)\/\(y\^2 - 1\) - 
      \(6\ \((32\ y\^5 - 32\ y\^3 + 6\ y)\)\ \(a(6)\)\ y\)\/\(y\^2 - 1\) - 
      \(8\ \((128\ y\^7 - 192\ y\^5 + 80\ y\^3 - 8\ y)\)\ \(a(8)\)\ y
          \)\/\(y\^2 - 1\) - 
      \(10\ \((512\ y\^9 - 1024\ y\^7 + 672\ y\^5 - 160\ y\^3 + 10\ y)\)\ 
          \(a(10)\)\ y\)\/\(y\^2 - 1\) - 
      \(12\ \((2048\ y\^11 - 5120\ y\^9 + 4608\ y\^7 - 1792\ y\^5 + 
              280\ y\^3 - 12\ y)\)\ \(a(12)\)\ y\)\/\(y\^2 - 1\) + a(0) + 
      \(4\ \((2\ y\^2 - 1)\)\ \(a(2)\)\)\/\(y\^2 - 1\) + 
      \((2\ y\^2 - 1)\)\ \(a(2)\) + 
      \(16\ \((8\ y\^4 - 8\ y\^2 + 1)\)\ \(a(4)\)\)\/\(y\^2 - 1\) + 
      \((8\ y\^4 - 8\ y\^2 + 1)\)\ \(a(4)\) + 
      \(36\ \((32\ y\^6 - 48\ y\^4 + 18\ y\^2 - 1)\)\ \(a(6)\)\)\/\(y\^2 - 1
          \) + \((32\ y\^6 - 48\ y\^4 + 18\ y\^2 - 1)\)\ \(a(6)\) + 
      \(64\ \((128\ y\^8 - 256\ y\^6 + 160\ y\^4 - 32\ y\^2 + 1)\)\ 
          \(a(8)\)\)\/\(y\^2 - 1\) + 
      \((128\ y\^8 - 256\ y\^6 + 160\ y\^4 - 32\ y\^2 + 1)\)\ \(a(8)\) + 
      \(100\ \((
            512\ y\^10 - 1280\ y\^8 + 1120\ y\^6 - 400\ y\^4 + 50\ y\^2 - 1)
            \)\ \(a(10)\)\)\/\(y\^2 - 1\) + 
      \((512\ y\^10 - 1280\ y\^8 + 1120\ y\^6 - 400\ y\^4 + 50\ y\^2 - 1)\)\ 
        \(a(10)\) + 
      \(144\ \((
            2048\ y\^12 - 6144\ y\^10 + 6912\ y\^8 - 3584\ y\^6 + 840\ y\^4 - 
              72\ y\^2 + 1)\)\ \(a(12)\)\)\/\(y\^2 - 1\) + 
      \((2048\ y\^12 - 6144\ y\^10 + 6912\ y\^8 - 3584\ y\^6 + 840\ y\^4 - 
            72\ y\^2 + 1)\)\ \(a(12)\)\)], "Output"],

Cell[TextData[{
  "Return to ",
  ButtonBox["conclusions",
    ButtonData:>"conclusions",
    ButtonStyle->"Hyperlink"]
}], "Text"],

Cell[BoxData[
    \(TraditionalForm
    \`\(-\(\(4\ \(a(2)\)\ y\^2\)\/\(y\^2 - 1\)\)\) - 
      \(4\ \((8\ y\^3 - 4\ y)\)\ \(a(4)\)\ y\)\/\(y\^2 - 1\) - 
      \(6\ \((32\ y\^5 - 32\ y\^3 + 6\ y)\)\ \(a(6)\)\ y\)\/\(y\^2 - 1\) - 
      \(8\ \((128\ y\^7 - 192\ y\^5 + 80\ y\^3 - 8\ y)\)\ \(a(8)\)\ y
          \)\/\(y\^2 - 1\) - 
      \(10\ \((512\ y\^9 - 1024\ y\^7 + 672\ y\^5 - 160\ y\^3 + 10\ y)\)\ 
          \(a(10)\)\ y\)\/\(y\^2 - 1\) - 
      \(12\ \((2048\ y\^11 - 5120\ y\^9 + 4608\ y\^7 - 1792\ y\^5 + 
              280\ y\^3 - 12\ y)\)\ \(a(12)\)\ y\)\/\(y\^2 - 1\) + a(0) + 
      \(4\ \((2\ y\^2 - 1)\)\ \(a(2)\)\)\/\(y\^2 - 1\) + 
      \((2\ y\^2 - 1)\)\ \(a(2)\) + 
      \(16\ \((8\ y\^4 - 8\ y\^2 + 1)\)\ \(a(4)\)\)\/\(y\^2 - 1\) + 
      \((8\ y\^4 - 8\ y\^2 + 1)\)\ \(a(4)\) + 
      \(36\ \((32\ y\^6 - 48\ y\^4 + 18\ y\^2 - 1)\)\ \(a(6)\)\)\/\(y\^2 - 1
          \) + \((32\ y\^6 - 48\ y\^4 + 18\ y\^2 - 1)\)\ \(a(6)\) + 
      \(64\ \((128\ y\^8 - 256\ y\^6 + 160\ y\^4 - 32\ y\^2 + 1)\)\ 
          \(a(8)\)\)\/\(y\^2 - 1\) + 
      \((128\ y\^8 - 256\ y\^6 + 160\ y\^4 - 32\ y\^2 + 1)\)\ \(a(8)\) + 
      \(100\ \((
            512\ y\^10 - 1280\ y\^8 + 1120\ y\^6 - 400\ y\^4 + 50\ y\^2 - 1)
            \)\ \(a(10)\)\)\/\(y\^2 - 1\) + 
      \((512\ y\^10 - 1280\ y\^8 + 1120\ y\^6 - 400\ y\^4 + 50\ y\^2 - 1)\)\ 
        \(a(10)\) + 
      \(144\ \((
            2048\ y\^12 - 6144\ y\^10 + 6912\ y\^8 - 3584\ y\^6 + 840\ y\^4 - 
              72\ y\^2 + 1)\)\ \(a(12)\)\)\/\(y\^2 - 1\) + 
      \((2048\ y\^12 - 6144\ y\^10 + 6912\ y\^8 - 3584\ y\^6 + 840\ y\^4 - 
            72\ y\^2 + 1)\)\ \(a(12)\)\)], "Output"]
}, Open  ]],

Cell["This might go quicker if we simplify it.", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp4 = Simplify[temp3]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`2048\ \(a(12)\)\ y\^12 + 512\ \(a(10)\)\ y\^10 + 
      264192\ \(a(12)\)\ y\^10 + 128\ \(a(8)\)\ y\^8 + 
      44800\ \(a(10)\)\ y\^8 - 546048\ \(a(12)\)\ y\^8 + 32\ \(a(6)\)\ y\^6 + 
      6912\ \(a(8)\)\ y\^6 - 70560\ \(a(10)\)\ y\^6 + 
      383488\ \(a(12)\)\ y\^6 + 8\ \(a(4)\)\ y\^4 + 912\ \(a(6)\)\ y\^4 - 
      7520\ \(a(8)\)\ y\^4 + 33200\ \(a(10)\)\ y\^4 - 
      106680\ \(a(12)\)\ y\^4 + 88\ \(a(4)\)\ y\^2 - 558\ \(a(6)\)\ y\^2 + 
      1888\ \(a(8)\)\ y\^2 - 4750\ \(a(10)\)\ y\^2 + 10008\ \(a(12)\)\ y\^2 + 
      a(0) + \((2\ y\^2 + 3)\)\ \(a(2)\) - 15\ \(a(4)\) + 35\ \(a(6)\) - 
      63\ \(a(8)\) + 99\ \(a(10)\) - 143\ \(a(12)\)\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "We need to find algebraic equations for the a[i]'s.  We will use the \
orthonality property of the polynomials to pick out equations for each a[i]. \
\n\n",
  Cell[BoxData[
      FormBox[
        RowBox[{\(\[Integral]\_\(-1\)\%1\), " ", 
          RowBox[{
            FormBox[\(\ \(\[Rho](y)\)*\(\(T\_j\)(y)\)\),
              "TraditionalForm"], "*", 
            RowBox[{"(", 
              RowBox[{
              "substituted", " ", "ODE", " ", "with", " ", "a", " ", 
                "number", " ", "of", " ", 
                RowBox[{
                  FormBox[\(\(T\_i\)(y)\),
                    "TraditionalForm"], "'"}], "s", 
                \(\[DifferentialD]y\)}]}]}]}], TraditionalForm]]],
  " . \n\nOne other \"trick\" needs to be recognized.  The orthogonality \
property will take care of \"choosing\" the nonzero pieces of the equation \
for each coefficient even if we multiply everthing out ahead of time as we \
have done above.\n\nNow set up integrals noting the complete tricks needed \
for orthogonality.  We multiply by the weighting function and an arbitrary \
CheybyshevT.  We will let the \"i\" take on all possible values between 0 and \
n to form n/2 equations for the n/2 coefficients.  We note that the \
integration will take of itself"
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  CellTags->"innerproduct"],

Cell[TextData[{
  "Return to ",
  ButtonBox["conclusions",
    ButtonData:>"conclusions",
    ButtonStyle->"Hyperlink"]
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp5 = 
      \(temp4\ 1\/\@\(1 - y\^2\)\ ChebyshevT[i, y]\ 2\)\/\(\[Pi]\ c[i]\)\)], 
  "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\(1\/\(\[Pi]\ \@\(1 - y\^2\)\ \(c(i)\)\)\((
      2\ \((2048\ \(a(12)\)\ y\^12 + 512\ \(a(10)\)\ y\^10 + 
            264192\ \(a(12)\)\ y\^10 + 128\ \(a(8)\)\ y\^8 + 
            44800\ \(a(10)\)\ y\^8 - 546048\ \(a(12)\)\ y\^8 + 
            32\ \(a(6)\)\ y\^6 + 6912\ \(a(8)\)\ y\^6 - 
            70560\ \(a(10)\)\ y\^6 + 383488\ \(a(12)\)\ y\^6 + 
            8\ \(a(4)\)\ y\^4 + 912\ \(a(6)\)\ y\^4 - 7520\ \(a(8)\)\ y\^4 + 
            33200\ \(a(10)\)\ y\^4 - 106680\ \(a(12)\)\ y\^4 + 
            88\ \(a(4)\)\ y\^2 - 558\ \(a(6)\)\ y\^2 + 1888\ \(a(8)\)\ y\^2 - 
            4750\ \(a(10)\)\ y\^2 + 10008\ \(a(12)\)\ y\^2 + a(0) + 
            \((2\ y\^2 + 3)\)\ \(a(2)\) - 15\ \(a(4)\) + 35\ \(a(6)\) - 
            63\ \(a(8)\) + 99\ \(a(10)\) - 143\ \(a(12)\))\)\ \(\(T\_i\)(y)\))
      \)\)\)], "Output"]
}, Open  ]],

Cell["\<\
Here we generate the n/2 equations in a table form.  The original \
equation was equal to 0, now each of these pieces is also equal to 0.  \
\>", 
  "Text"],

Cell[BoxData[
    \(\(temp6 = Table[temp5, {i, 0, n, 2}]; \)\)], "Input",
  AspectRatioFixed->True],

Cell["Let's get rid of the c[i]'s", "Text"],

Cell["temp7=(temp6/.c[0]->2)/.Table[c[i]->1,{i,2,n,2}];", "Input"],

Cell[TextData[{
  "At this point, we could just integrate \"temp7\" from -1 to 1 and we would \
have the desired algebraic equations.  However the integration is very slow \
for a lot of terms and we need to speed it up.  To do this, we factor out the \
common terms.  These are all of the form ",
  Cell[BoxData[
      \(\(\ y\^\(2  n\)\)\)]],
  "/ \[Sqrt](1-",
  Cell[BoxData[
      \(\(y\^2)\)\)]],
  ".  We first collect the terms in powers of y.  We use only the numerator \
for this so that the ",
  StyleBox["collect",
    FontSlant->"Italic"],
  " works. "
}], "Text"],

Cell["temp8=Collect[Expand[Numerator[temp7]],y];", "Input"],

Cell[TextData[{
  "Here is where we factor them out.  The Transpose is needed to put the \
matrix back in the form that would multiply a column vector of the  ",
  Cell[BoxData[
      \(\(\ y\^\(2  n\)\)\)]],
  "/ \[Sqrt](1-",
  Cell[BoxData[
      \(\(y\^2)\)\)]],
  "'s.   "
}], "Text"],

Cell[CellGroupData[{

Cell["temp9=Transpose[Table[Coefficient[temp8,y,i],{i,0,2 n,2}]]", "Input"],

Cell[BoxData[
    FormBox[
      RowBox[{"(", GridBox[{
            {
              \(a(0) + 3\ \(a(2)\) - 15\ \(a(4)\) + 35\ \(a(6)\) - 
                63\ \(a(8)\) + 99\ \(a(10)\) - 143\ \(a(12)\)\), 
              \(2\ \(a(2)\) + 88\ \(a(4)\) - 558\ \(a(6)\) + 1888\ \(a(8)\) - 
                4750\ \(a(10)\) + 10008\ \(a(12)\)\), 
              \(8\ \(a(4)\) + 912\ \(a(6)\) - 7520\ \(a(8)\) + 
                33200\ \(a(10)\) - 106680\ \(a(12)\)\), 
              \(32\ \(a(6)\) + 6912\ \(a(8)\) - 70560\ \(a(10)\) + 
                383488\ \(a(12)\)\), 
              \(128\ \(a(8)\) + 44800\ \(a(10)\) - 546048\ \(a(12)\)\), 
              \(512\ \(a(10)\) + 264192\ \(a(12)\)\), \(2048\ \(a(12)\)\), 
              "0", "0", "0", "0", "0", "0"},
            {
              \(\(-2\)\ \(a(0)\) - 6\ \(a(2)\) + 30\ \(a(4)\) - 
                70\ \(a(6)\) + 126\ \(a(8)\) - 198\ \(a(10)\) + 
                286\ \(a(12)\)\), 
              \(4\ \(a(0)\) + 8\ \(a(2)\) - 236\ \(a(4)\) + 1256\ \(a(6)\) - 
                4028\ \(a(8)\) + 9896\ \(a(10)\) - 20588\ \(a(12)\)\), 
              \(8\ \(a(2)\) + 336\ \(a(4)\) - 4056\ \(a(6)\) + 
                22592\ \(a(8)\) - 85400\ \(a(10)\) + 253392\ \(a(12)\)\), 
              \(32\ \(a(4)\) + 3584\ \(a(6)\) - 43904\ \(a(8)\) + 
                273920\ \(a(10)\) - 1193696\ \(a(12)\)\), 
              \(128\ \(a(6)\) + 27392\ \(a(8)\) - 371840\ \(a(10)\) + 
                2626048\ \(a(12)\)\), 
              \(512\ \(a(8)\) + 178176\ \(a(10)\) - 2712576\ \(a(12)\)\), 
              \(2048\ \(a(10)\) + 1052672\ \(a(12)\)\), \(8192\ \(a(12)\)\), 
              "0", "0", "0", "0", "0"},
            {
              \(2\ \(a(0)\) + 6\ \(a(2)\) - 30\ \(a(4)\) + 70\ \(a(6)\) - 
                126\ \(a(8)\) + 198\ \(a(10)\) - 286\ \(a(12)\)\), 
              \(\(-16\)\ \(a(0)\) - 44\ \(a(2)\) + 416\ \(a(4)\) - 
                1676\ \(a(6)\) + 4784\ \(a(8)\) - 11084\ \(a(10)\) + 
                22304\ \(a(12)\)\), 
              \(16\ \(a(0)\) + 16\ \(a(2)\) - 1632\ \(a(4)\) + 
                11312\ \(a(6)\) - 46256\ \(a(8)\) + 143984\ \(a(10)\) - 
                375776\ \(a(12)\)\), 
              \(32\ \(a(2)\) + 1280\ \(a(4)\) - 23456\ \(a(6)\) + 
                164352\ \(a(8)\) - 748320\ \(a(10)\) + 2633984\ \(a(12)\)\), 
              \(128\ \(a(4)\) + 14080\ \(a(6)\) - 230656\ \(a(8)\) + 
                1749760\ \(a(10)\) - 8934784\ \(a(12)\)\), 
              \(512\ \(a(6)\) + 108544\ \(a(8)\) - 1844736\ \(a(10)\) + 
                15400960\ \(a(12)\)\), 
              \(2048\ \(a(8)\) + 708608\ \(a(10)\) - 12959744\ \(a(12)\)\), 
              \(8192\ \(a(10)\) + 4194304\ \(a(12)\)\), \(32768\ \(a(12)\)\), 
              "0", "0", "0", "0"},
            {
              \(\(-2\)\ \(a(0)\) - 6\ \(a(2)\) + 30\ \(a(4)\) - 
                70\ \(a(6)\) + 126\ \(a(8)\) - 198\ \(a(10)\) + 
                286\ \(a(12)\)\), 
              \(36\ \(a(0)\) + 104\ \(a(2)\) - 716\ \(a(4)\) + 
                2376\ \(a(6)\) - 6044\ \(a(8)\) + 13064\ \(a(10)\) - 
                25164\ \(a(12)\)\), 
              \(\(-96\)\ \(a(0)\) - 216\ \(a(2)\) + 4592\ \(a(4)\) - 
                25272\ \(a(6)\) + 89056\ \(a(8)\) - 246904\ \(a(10)\) + 
                587376\ \(a(12)\)\), 
              \(64\ \(a(0)\) - 9120\ \(a(4)\) + 88576\ \(a(6)\) - 
                469824\ \(a(8)\) + 1798656\ \(a(10)\) - 5577376\ \(a(12)\)\), 
              \(128\ \(a(2)\) + 4864\ \(a(4)\) - 122112\ \(a(6)\) + 
                1091328\ \(a(8)\) - 6120960\ \(a(10)\) + 
                25779456\ \(a(12)\)\), 
              \(512\ \(a(4)\) + 55296\ \(a(6)\) - 1140224\ \(a(8)\) + 
                10510336\ \(a(10)\) - 63828480\ \(a(12)\)\), 
              \(2048\ \(a(6)\) + 430080\ \(a(8)\) - 8798208\ \(a(10)\) + 
                86470656\ \(a(12)\)\), 
              \(8192\ \(a(8)\) + 2818048\ \(a(10)\) - 60235776\ \(a(12)\)\), 
              \(32768\ \(a(10)\) + 16711680\ \(a(12)\)\), 
              \(131072\ \(a(12)\)\), "0", "0", "0"},
            {
              \(2\ \(a(0)\) + 6\ \(a(2)\) - 30\ \(a(4)\) + 70\ \(a(6)\) - 
                126\ \(a(8)\) + 198\ \(a(10)\) - 286\ \(a(12)\)\), 
              \(\(-64\)\ \(a(0)\) - 188\ \(a(2)\) + 1136\ \(a(4)\) - 
                3356\ \(a(6)\) + 7808\ \(a(8)\) - 15836\ \(a(10)\) + 
                29168\ \(a(12)\)\), 
              \(320\ \(a(0)\) + 832\ \(a(2)\) - 10416\ \(a(4)\) + 
                48736\ \(a(6)\) - 156032\ \(a(8)\) + 402080\ \(a(10)\) - 
                899632\ \(a(12)\)\), 
              \(\(-512\)\ \(a(0)\) - 896\ \(a(2)\) + 35328\ \(a(4)\) - 
                254784\ \(a(6)\) + 1131520\ \(a(8)\) - 3836608\ \(a(10)\) + 
                10870272\ \(a(12)\)\), 
              \(256\ \(a(0)\) - 256\ \(a(2)\) - 46336\ \(a(4)\) + 
                584448\ \(a(6)\) - 3831296\ \(a(8)\) + 17686784\ \(a(10)\) - 
                64933632\ \(a(12)\)\), 
              \(512\ \(a(2)\) + 18432\ \(a(4)\) - 599552\ \(a(6)\) + 
                6537216\ \(a(8)\) - 43659776\ \(a(10)\) + 
                215373824\ \(a(12)\)\), 
              \(2048\ \(a(4)\) + 217088\ \(a(6)\) - 5423104\ \(a(8)\) + 
                58929152\ \(a(10)\) - 415295488\ \(a(12)\)\), 
              \(8192\ \(a(6)\) + 1703936\ \(a(8)\) - 40837120\ \(a(10)\) + 
                462159872\ \(a(12)\)\), 
              \(32768\ \(a(8)\) + 11206656\ \(a(10)\) - 
                274399232\ \(a(12)\)\), 
              \(131072\ \(a(10)\) + 66584576\ \(a(12)\)\), 
              \(524288\ \(a(12)\)\), "0", "0"},
            {
              \(\(-2\)\ \(a(0)\) - 6\ \(a(2)\) + 30\ \(a(4)\) - 
                70\ \(a(6)\) + 126\ \(a(8)\) - 198\ \(a(10)\) + 
                286\ \(a(12)\)\), 
              \(100\ \(a(0)\) + 296\ \(a(2)\) - 1676\ \(a(4)\) + 
                4616\ \(a(6)\) - 10076\ \(a(8)\) + 19400\ \(a(10)\) - 
                34316\ \(a(12)\)\), 
              \(\(-800\)\ \(a(0)\) - 2200\ \(a(2)\) + 20784\ \(a(4)\) - 
                85624\ \(a(6)\) + 254240\ \(a(8)\) - 620600\ \(a(10)\) + 
                1328560\ \(a(12)\)\), 
              \(2240\ \(a(0)\) + 5120\ \(a(2)\) - 103200\ \(a(4)\) + 
                615936\ \(a(6)\) - 2417344\ \(a(8)\) + 7482880\ \(a(10)\) - 
                19761696\ \(a(12)\)\), 
              \(\(-2560\)\ \(a(0)\) - 3200\ \(a(2)\) + 229120\ \(a(4)\) - 
                2065920\ \(a(6)\) + 11097344\ \(a(8)\) - 
                44599040\ \(a(10)\) + 147568896\ \(a(12)\)\), 
              \(1024\ \(a(0)\) - 2048\ \(a(2)\) - 222720\ \(a(4)\) + 
                3481600\ \(a(6)\) - 27259392\ \(a(8)\) + 
                147556352\ \(a(10)\) - 626653696\ \(a(12)\)\), 
              \(2048\ \(a(2)\) + 69632\ \(a(4)\) - 2834432\ \(a(6)\) + 
                36564992\ \(a(8)\) - 283699200\ \(a(10)\) + 
                1605615616\ \(a(12)\)\), 
              \(8192\ \(a(4)\) + 851968\ \(a(6)\) - 25108480\ \(a(8)\) + 
                314572800\ \(a(10)\) - 2525265920\ \(a(12)\)\), 
              \(32768\ \(a(6)\) + 6750208\ \(a(8)\) - 185794560\ \(a(10)\) + 
                2380726272\ \(a(12)\)\), 
              \(131072\ \(a(8)\) + 44564480\ \(a(10)\) - 
                1230897152\ \(a(12)\)\), 
              \(524288\ \(a(10)\) + 265289728\ \(a(12)\)\), 
              \(2097152\ \(a(12)\)\), "0"},
            {
              \(2\ \(a(0)\) + 6\ \(a(2)\) - 30\ \(a(4)\) + 70\ \(a(6)\) - 
                126\ \(a(8)\) + 198\ \(a(10)\) - 286\ \(a(12)\)\), 
              \(\(-144\)\ \(a(0)\) - 428\ \(a(2)\) + 2336\ \(a(4)\) - 
                6156\ \(a(6)\) + 12848\ \(a(8)\) - 23756\ \(a(10)\) + 
                40608\ \(a(12)\)\), 
              \(1680\ \(a(0)\) + 4752\ \(a(2)\) - 37856\ \(a(4)\) + 
                140976\ \(a(6)\) - 392752\ \(a(8)\) + 916720\ \(a(10)\) - 
                1894752\ \(a(12)\)\), 
              \(\(-7168\)\ \(a(0)\) - 18144\ \(a(2)\) + 254208\ \(a(4)\) - 
                1319584\ \(a(6)\) + 4720128\ \(a(8)\) - 13611552\ \(a(10)\) + 
                33967360\ \(a(12)\)\), 
              \(13824\ \(a(0)\) + 27136\ \(a(2)\) - 824704\ \(a(4)\) + 
                6011136\ \(a(6)\) - 28032768\ \(a(8)\) + 
                101442816\ \(a(10)\) - 309250944\ \(a(12)\)\), 
              \(\(-12288\)\ \(a(0)\) - 9216\ \(a(2)\) + 1343488\ \(a(4)\) - 
                14627328\ \(a(6)\) + 92370944\ \(a(8)\) - 
                429849088\ \(a(10)\) + 1628209152\ \(a(12)\)\), 
              \(4096\ \(a(0)\) - 12288\ \(a(2)\) - 1032192\ \(a(4)\) + 
                19378176\ \(a(6)\) - 176744448\ \(a(8)\) + 
                1098694656\ \(a(10)\) - 5302550528\ \(a(12)\)\), 
              \(8192\ \(a(2)\) + 262144\ \(a(4)\) - 13049856\ \(a(6)\) + 
                194772992\ \(a(8)\) - 1723105280\ \(a(10)\) + 
                11010834432\ \(a(12)\)\), 
              \(32768\ \(a(4)\) + 3342336\ \(a(6)\) - 113967104\ \(a(8)\) + 
                1618673664\ \(a(10)\) - 14588116992\ \(a(12)\)\), 
              \(131072\ \(a(6)\) + 26738688\ \(a(8)\) - 
                832438272\ \(a(10)\) + 11918114816\ \(a(12)\)\), 
              \(524288\ \(a(8)\) + 177209344\ \(a(10)\) - 
                5454692352\ \(a(12)\)\), 
              \(2097152\ \(a(10)\) + 1056964608\ \(a(12)\)\), 
              \(8388608\ \(a(12)\)\)}
            }], ")"}], TraditionalForm]], "Output"]
}, Open  ]],

Cell["Here is what we integrate:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp10 = Table[y\^i\/\(\[Pi]\ \@\(1 - y\^2\)\), {i, 0, 2\ n, 2}]\)], 
  "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`{1\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^2\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^4\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^6\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^8\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^10\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^12\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^14\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^16\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^18\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^20\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^22\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^24\/\(\[Pi]\ \@\(1 - y\^2\)\)}\)], "Output"]
}, Open  ]],

Cell["Here is the integration showing the time it takes.  ", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Timing[temp11 = \[Integral]\_\(-1\)\%1 temp10 \[DifferentialD]y]\)], 
  "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`{3.29999999999999893`\ Second, {1, 1\/2, 3\/8, 5\/16, 35\/128, 63\/256, 
        231\/1024, 429\/2048, 6435\/32768, 12155\/65536, 46189\/262144, 
        88179\/524288, 676039\/4194304}}\)], "Output"]
}, Open  ]],

Cell["\<\
Now multiply the coefficients times the integrated polynomial \
pieces.   \
\>", "Text"],

Cell[CellGroupData[{

Cell["temp12= Simplify[Expand[temp9.temp11]] ", "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`{a(0) + 
        4\ \((a(2) + 8\ \(a(4)\) + 27\ \(a(6)\) + 64\ \(a(8)\) + 
              125\ \(a(10)\) + 216\ \(a(12)\))\), 
      a(2) + 48\ 
          \((a(4) + 4\ \(a(6)\) + 10\ \(a(8)\) + 20\ \(a(10)\) + 
              35\ \(a(12)\))\), 
      a(4) + 24\ 
          \((5\ \(a(6)\) + 16\ \(a(8)\) + 35\ \(a(10)\) + 64\ \(a(12)\))\), 
      a(6) + 16\ \((14\ \(a(8)\) + 40\ \(a(10)\) + 81\ \(a(12)\))\), 
      a(8) + 360\ \(a(10)\) + 960\ \(a(12)\), a(10) + 528\ \(a(12)\), a(12)}
      \)], "Output"]
}, Open  ]],

Cell["\<\
Each of the entries in this table is an equation that equals 0.  We \
can solve this at this point at get the trivial solution since there are n/2 \
unknown a(j)'s and n/2 equations.  Of course, since we have thus far just \
substituted into a homogeneous equation and not yet used the nonhomogeneous \
boundary conditions, what other answer could we expect!  

So we need to enforce the nonhomogeneous boundary conditions both to get a \
nonzero answer and also to get the correct answer.  

As mentioned above, we will replace one of these n/2+1 equations with the \
boundary condition.  You might ask which one should we replace.  The obvious \
one(s) to replace are the equations generated by the highest values of i.  We \
justify this two ways.  One is that we employ extra terms, which are \
necessarily at high values of \"i\" in the expansion so that there are enough \
to get a good answer.  Thus dropping a few at high i does not seem to hurt.  \
A second way to think of this is that the functions have an intrinsic \
\"frequency\" and we are just dropping off some of the higher frequencies \
that are not important. 

Since the solution is symmetric we have already implicitly made the two \
boundary conditions equal by using only even polynomials.  Thus need to drop \
off only one the last one.   Note that on the boundaries, each polynomial has \
a value of 1 thus we write the boundary condition as  \
\>", "Text",
  CellTags->"boundaryc"],

Cell[TextData[{
  "Return to ",
  ButtonBox["conclusions",
    ButtonData:>"conclusions",
    ButtonStyle->"Hyperlink"]
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{
      RowBox[{"bc1", "=", " ", 
        RowBox[{
          UnderoverscriptBox["\[Sum]", GridBox[{
                {\(i = 0\)},
                {\(\[CapitalDelta]\[MediumSpace]i = 2\)}
                }], "n"], \(a[i]\)}]}], " "}]], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`a(0) + a(2) + a(4) + a(6) + a(8) + a(10) + a(12)\)], 
  "Output"]
}, Open  ]],

Cell["\<\
Here is where we take the first n/2 rows of the coefficients from \
the ODE and Append the one boundary condition.  Note how the value of the \
boundary condition was included while allowing \"LogicalExpand\" to do its \
trick.  This simple command produces a set of equations that can be solved.   \
 \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp13 = 
      LogicalExpand[
        Append[Table[
              temp12\[LeftDoubleBracket]i\[RightDoubleBracket], {i, 1, 
                n\/2}], bc1 - 1] == 0]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`a(0) + a(2) + a(4) + a(6) + a(8) + a(10) + a(12) - 1 == 0 \[And] 
      a(10) + 528\ \(a(12)\) == 0 \[And] 
      a(8) + 360\ \(a(10)\) + 960\ \(a(12)\) == 0 \[And] 
      a(2) + 48\ 
            \((a(4) + 4\ \(a(6)\) + 10\ \(a(8)\) + 20\ \(a(10)\) + 
                35\ \(a(12)\))\) == 0 \[And] 
      a(4) + 24\ 
            \((5\ \(a(6)\) + 16\ \(a(8)\) + 35\ \(a(10)\) + 64\ \(a(12)\))
              \) == 0 \[And] 
      a(6) + 16\ \((14\ \(a(8)\) + 40\ \(a(10)\) + 81\ \(a(12)\))\) == 0 
      \[And] a(0) + 
          4\ \((a(2) + 8\ \(a(4)\) + 27\ \(a(6)\) + 64\ \(a(8)\) + 
                125\ \(a(10)\) + 216\ \(a(12)\))\) == 0\)], "Output"]
}, Open  ]],

Cell[TextData["We solve for the coefficients."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(coef = Solve[temp13, Table[a[i], {i, 0, n, 2}]]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`{{a(0) \[Rule] 767930908896\/542232220657, 
        a(2) \[Rule] \(-\(230627821200\/542232220657\)\), 
        a(4) \[Rule] 4970970624\/542232220657, 
        a(6) \[Rule] \(-\(42026256\/542232220657\)\), 
        a(8) \[Rule] 189120\/542232220657, 
        a(10) \[Rule] \(-\(528\/542232220657\)\), 
        a(12) \[Rule] 1\/542232220657}}\)], "Output"]
}, Open  ]],

Cell["We want to reconstruct the solution.  First make the series.", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{"expand1", "=", 
      RowBox[{
        UnderoverscriptBox["\[Sum]", GridBox[{
              {\(i = 0\)},
              {\(\[CapitalDelta]\[MediumSpace]i = 2\)}
              }], "n"], \(a[i]\ ChebyshevT[i, y]\)}]}]], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`a(0) + \((2\ y\^2 - 1)\)\ \(a(2)\) + 
      \((8\ y\^4 - 8\ y\^2 + 1)\)\ \(a(4)\) + 
      \((32\ y\^6 - 48\ y\^4 + 18\ y\^2 - 1)\)\ \(a(6)\) + 
      \((128\ y\^8 - 256\ y\^6 + 160\ y\^4 - 32\ y\^2 + 1)\)\ \(a(8)\) + 
      \((512\ y\^10 - 1280\ y\^8 + 1120\ y\^6 - 400\ y\^4 + 50\ y\^2 - 1)\)\ 
        \(a(10)\) + 
      \((2048\ y\^12 - 6144\ y\^10 + 6912\ y\^8 - 3584\ y\^6 + 840\ y\^4 - 
            72\ y\^2 + 1)\)\ \(a(12)\)\)], "Output"]
}, Open  ]],

Cell["\<\
Then substitute the coefficients that we calculated above to get \
the answer.  \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(anscheby = 
      expand1 /. coef\[LeftDoubleBracket]1\[RightDoubleBracket]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\(-\(\(230627821200\ \((2\ y\^2 - 1)\)\)\/542232220657\)\) + 
      \(4970970624\ \((8\ y\^4 - 8\ y\^2 + 1)\)\)\/542232220657 - 
      \(42026256\ \((32\ y\^6 - 48\ y\^4 + 18\ y\^2 - 1)\)\)\/542232220657 + 
      \(189120\ \((128\ y\^8 - 256\ y\^6 + 160\ y\^4 - 32\ y\^2 + 1)
            \)\)\/542232220657 - 
      \(528\ \((
            512\ y\^10 - 1280\ y\^8 + 1120\ y\^6 - 400\ y\^4 + 50\ y\^2 - 1)
            \)\)\/542232220657 + 
      \(2048\ y\^12 - 6144\ y\^10 + 6912\ y\^8 - 3584\ y\^6 + 840\ y\^4 - 
          72\ y\^2 + 1\)\/542232220657 + 767930908896\/542232220657\)], 
  "Output"]
}, Open  ]],

Cell["Here is a plot of the numerical solution.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(p2 = Plot[anscheby, {y, \(-1\), 1}]; \)\)], "Input",
  AspectRatioFixed->True],

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              PrintPrecision->5], " ", \(y\^12\)}], "+", 
          InterpretationBox[\(O(y\^13)\),
            SeriesData[ y, 0, {}, 0, 13, 1]]}],
        SeriesData[ 
        y, 0, {1.8508157176809119, 0, -0.9254078588395338, 0, 
          0.077117321558895779, 0, -0.0025705773336581338, 0, 
          4.5903048641856247*^-05, 0, -5.0989223706588443*^-07, 0, 
          3.7769795338213661*^-09}, 0, 13, 1]], TraditionalForm]], "Output"]
}, Open  ]],

Cell["Here is the difference.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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  AspectRatioFixed->True],

Cell[BoxData[
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      InterpretationBox[
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        "1.35447209004269097`*^-14", "-", 
          \(9.28923604703868477`*^-13\ y\^2\), "+", 
          \(1.11427811422259992`*^-11\ y\^4\), "-", 
          \(5.20098183270922831`*^-11\ y\^6\), "+", 
          \(1.18959357182848046`*^-10\ y\^8\), "-", 
          \(1.42958503153647642`*^-10\ y\^10\), "+", 
          \(8.69234629137714698`*^-11\ y\^12\), "+", 
          InterpretationBox[\(O(y\^13)\),
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Cell[TextData[{
  "Here is an integral measure of the error.  If we use enough terms, it is \
difficult for ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to compute the error.  "
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(e[n] = NIntegrate[\((soln1 - anscheby)\), {y, \(-1\), 1}]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`5.91919878400052912`*^-16\)], "Output"]
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Cell[CellGroupData[{

Cell["\<\
Here are some answers from previous expansions and we see that the \
answer is getting better really quick.\
\>", "Text",
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  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(e[2]\)], "Input",
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Cell[BoxData[
    \(TraditionalForm\`0.225926560420916233`\)], "Output",
  CellLabelAutoDelete->True]
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Cell[CellGroupData[{

Cell[BoxData[
    \(e[4]\)], "Input",
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  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`0.000951142525144034678`\)], "Output",
  CellLabelAutoDelete->True]
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Cell[CellGroupData[{

Cell["e[6]", "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True],

Cell[BoxData[
    \(TraditionalForm\`\(-6.18435756238768874`*^-7\)\)], "Output",
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Cell[CellGroupData[{

Cell[BoxData[
    \(e[8]\)], "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`8.94718624911067372`*^-10\)], "Output",
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Cell[CellGroupData[{

Cell[BoxData[
    \(e[10]\)], "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`\(-9.80993079362783149`*^-13\)\)], "Output",
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Cell[CellGroupData[{

Cell[BoxData[
    \(e[12]\)], "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`1.11766206719230254`*^-15\)], "Output",
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Cell[CellGroupData[{

Cell[TextData["Numerical solution using just a power series"], "Subsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
Now solve the equation by assuming that the solution is just a \
power series.  This will be substituted into the ODE and we will generate \
algebraic equations.  We will equate the coefficients for each power of y to \
0 to generate algebraic equations.\
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["Here, again, is the ODE.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(ode1 = \[PartialD]\_{y, 2}vv[y]\  + \ vv[y]\)], "Input",
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Cell[BoxData[
    FormBox[
      RowBox[{\(vv(y)\), "+", 
        RowBox[{
          SuperscriptBox["vv", "\[DoublePrime]",
            MultilineFunction->None], "(", "y", ")"}]}], TraditionalForm]], 
  "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(n = 12; \)\)], "Input",
  AspectRatioFixed->True],

Cell["\<\
Here is our presumed solution.  Note that it is only even powers \
because we expect it to be symmetric.  \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{"seriesexpand", "=", 
      RowBox[{
        UnderoverscriptBox["\[Sum]", GridBox[{
              {\(i = 0\)},
              {\(\[CapitalDelta]\[MediumSpace]i = 2\)}
              }], "n"], \(a[i]\ y\^i\)}]}]], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\(a(12)\)\ y\^12 + \(a(10)\)\ y\^10 + \(a(8)\)\ y\^8 + \(a(6)\)\ y\^6 + 
      \(a(4)\)\ y\^4 + \(a(2)\)\ y\^2 + a(0)\)], "Output"]
}, Open  ]],

Cell["Here we substitute.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp1 = 
      ode1 /. {vv[y] \[Rule] seriesexpand, 
          \[PartialD]\_{y, a1_}vv[y] \[Rule] \[PartialD]\_{y, a1}seriesexpand}
          \)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\(a(12)\)\ y\^12 + \(a(10)\)\ y\^10 + 132\ \(a(12)\)\ y\^10 + 
      \(a(8)\)\ y\^8 + 90\ \(a(10)\)\ y\^8 + \(a(6)\)\ y\^6 + 
      56\ \(a(8)\)\ y\^6 + \(a(4)\)\ y\^4 + 30\ \(a(6)\)\ y\^4 + 
      \(a(2)\)\ y\^2 + 12\ \(a(4)\)\ y\^2 + a(0) + 2\ \(a(2)\)\)], "Output"]
}, Open  ]],

Cell["Expand so that we can collect the individual powers of y.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp2 = ExpandAll[temp1]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\(a(12)\)\ y\^12 + \(a(10)\)\ y\^10 + 132\ \(a(12)\)\ y\^10 + 
      \(a(8)\)\ y\^8 + 90\ \(a(10)\)\ y\^8 + \(a(6)\)\ y\^6 + 
      56\ \(a(8)\)\ y\^6 + \(a(4)\)\ y\^4 + 30\ \(a(6)\)\ y\^4 + 
      \(a(2)\)\ y\^2 + 12\ \(a(4)\)\ y\^2 + a(0) + 2\ \(a(2)\)\)], "Output"]
}, Open  ]],

Cell["Here we collect.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp3 = Collect[Expand[temp2], y]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\(a(12)\)\ y\^12 + \((a(10) + 132\ \(a(12)\))\)\ y\^10 + 
      \((a(8) + 90\ \(a(10)\))\)\ y\^8 + \((a(6) + 56\ \(a(8)\))\)\ y\^6 + 
      \((a(4) + 30\ \(a(6)\))\)\ y\^4 + \((a(2) + 12\ \(a(4)\))\)\ y\^2 + 
      a(0) + 2\ \(a(2)\)\)], "Output"]
}, Open  ]],

Cell["This generates a table of the coefficients. ", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(eqs = Table[Coefficient[temp3, y, i], {i, 0, n, 2}]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`{a(0) + 2\ \(a(2)\), a(2) + 12\ \(a(4)\), a(4) + 30\ \(a(6)\), 
      a(6) + 56\ \(a(8)\), a(8) + 90\ \(a(10)\), a(10) + 132\ \(a(12)\), 
      a(12)}\)], "Output"]
}, Open  ]],

Cell["We need one boundary condition as above.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(bcx1 = seriesexpand /. y \[Rule] 1\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`a(0) + a(2) + a(4) + a(6) + a(8) + a(10) + a(12)\)], 
  "Output"]
}, Open  ]],

Cell["\<\
This is how to make the equations for the coefficients.  Note that \
we replace the highest power with the boundary condition. \
\>", "Text",
  CellTags->"powers"],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp10 = 
      LogicalExpand[
        Append[Table[
              eqs\[LeftDoubleBracket]i\[RightDoubleBracket], {i, 1, n/2}], 
            bcx1 - 1] == 0]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`a(0) + 2\ \(a(2)\) == 0 \[And] a(2) + 12\ \(a(4)\) == 0 \[And] 
      a(4) + 30\ \(a(6)\) == 0 \[And] a(6) + 56\ \(a(8)\) == 0 \[And] 
      a(8) + 90\ \(a(10)\) == 0 \[And] 
      a(0) + a(2) + a(4) + a(6) + a(8) + a(10) + a(12) - 1 == 0 \[And] 
      a(10) + 132\ \(a(12)\) == 0\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Return to ",
  ButtonBox["conclusions",
    ButtonData:>"conclusions",
    ButtonStyle->"Hyperlink"]
}], "Text"],

Cell[TextData["We solve for the coefficients."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(coef = Solve[temp10, Table[a[i], {i, 0, n, 2}]]\)], "Input",
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Cell[BoxData[
    \(TraditionalForm
    \`{{a(0) \[Rule] 479001600\/258805669, 
        a(2) \[Rule] \(-\(239500800\/258805669\)\), 
        a(4) \[Rule] 19958400\/258805669, 
        a(6) \[Rule] \(-\(665280\/258805669\)\), 
        a(8) \[Rule] 11880\/258805669, a(10) \[Rule] \(-\(132\/258805669\)\), 
        a(12) \[Rule] 1\/258805669}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData["We want to reconstruct the solution."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(ansx = 
      seriesexpand /. coef\[LeftDoubleBracket]1\[RightDoubleBracket]\)], 
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Cell[BoxData[
    \(TraditionalForm
    \`y\^12\/258805669 - \(132\ y\^10\)\/258805669 + 
      \(11880\ y\^8\)\/258805669 - \(665280\ y\^6\)\/258805669 + 
      \(19958400\ y\^4\)\/258805669 - \(239500800\ y\^2\)\/258805669 + 
      479001600\/258805669\)], "Output"]
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Cell[CellGroupData[{

Cell["Look at the series expansions.  ", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
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            StyleBoxAutoDelete->True,
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            StyleBox["0.0771173215700385661`",
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Cell[BoxData[
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Cell[BoxData[
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        "3.91298105029136422`*^-11", "-", 
          \(1.95649052514568211`*^-11\ y\^2\), "+", 
          \(1.63041802281327363`*^-12\ y\^4\), "-", 
          \(5.43471517788773894`*^-14\ y\^6\), "+", 
          \(9.70482917118930998`*^-16\ y\^8\), "-", 
          \(1.07831529350208699`*^-17\ y\^10\), "+", 
          \(8.16907029345119184`*^-20\ y\^12\), "+", 
          InterpretationBox[\(O(y\^13)\),
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Cell[CellGroupData[{

Cell["Here is an integral measure of the error.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(ep[n] = NIntegrate[\((soln1 - ansx)\), {y, \(-1\), 1}]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`6.30329954060986352`*^-11\)], "Output"]
}, Open  ]]
}, Closed]]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Here is a comparison of error for the Chebyshev and power series \
solutions.\
\>", "Subsection"],

Cell[CellGroupData[{

Cell["Here are some answers from the Chebyshev spectral solutions", "Text",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(e[2]\)], "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`0.225926560420916233`\)], "Output",
  CellLabelAutoDelete->True]
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Cell[CellGroupData[{

Cell[BoxData[
    \(e[4]\)], "Input",
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  CellLabelAutoDelete->True,
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Cell[BoxData[
    \(TraditionalForm\`0.000951142525144034678`\)], "Output",
  CellLabelAutoDelete->True]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(e[6]\)], "Input",
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  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`\(-6.18435756238768874`*^-7\)\)], "Output",
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Cell[CellGroupData[{

Cell[BoxData[
    \(e[8]\)], "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`8.94718624911067372`*^-10\)], "Output",
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Cell[CellGroupData[{

Cell[BoxData[
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  Evaluatable->False,
  CellLabelAutoDelete->True,
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    \(TraditionalForm\`\(-9.80993079362783149`*^-13\)\)], "Output",
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Cell[CellGroupData[{

Cell["Here are the power series solutions", "Text",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(ep[2]\)], "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`\(-0.218517884023528408`\)\)], "Output",
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Cell[CellGroupData[{

Cell["ep[4]", "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True],

Cell[BoxData[
    \(TraditionalForm\`0.00712314161749707253`\)], "Output",
  CellLabelAutoDelete->True]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(ep[6]\)], "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`\(-0.000131300598384822508`\)\)], "Output",
  CellLabelAutoDelete->True]
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Cell[CellGroupData[{

Cell[BoxData[
    \(ep[8]\)], "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`1.48455321892871117`*^-6\)], "Output",
  CellLabelAutoDelete->True]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(ep[10]\)], "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`\(-1.13778646358869672`*^-8\)\)], "Output",
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(ep[12]\)], "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`6.30335491890489763`*^-11\)], "Output",
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}, Open  ]]
}, Open  ]],

Cell["\<\
Here are the two methods compared, first the Spectral \
approach:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(testtable = {{2, 0.225926560420915478`}, {4, 
          0.000951142525143355577`}, {6, \(-6.18435756960661642`*^-7\)}, {8, 
          8.9471794231341768`*^-10}, {10, \(-9.81661958634065534`*^-13\)}, {
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            {"8", "8.9471794231341768`*^-10"},
            {"10", \(-9.81661958634065534`*^-13\)},
            {"12", "5.91919878400052912`*^-16"},
            {"14", "9.66027008208424398`*^-17"},
            {"16", "1.04611309920646089`*^-17"}
            },
          ColumnAlignments->{Decimal}], ")"}], TraditionalForm]], "Output"]
}, Open  ]],

Cell["Here is the power series approach,", "Text",
  CellTags->"comparison"],

Cell[TextData[{
  "Return to ",
  ButtonBox["conclusions",
    ButtonData:>"conclusions",
    ButtonStyle->"Hyperlink"]
}], "Text"],

Cell[BoxData[
    \(testpowertable = {{2, \(-0.218517884023529118`\)}, {4, 
          0.00712314161749632557`}, {6, \(-0.00013130059838550192`\)}, {8, 
          1.4845532182493013`*^-6}, {10, \(-1.13778653577798039`*^-8\)}, {12, 
          6.30329954060986352`*^-11}}\)], "Input"],

Cell[BoxData[
    FormBox[
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            {"2", \(-0.218517884023529118`\)},
            {"4", "0.00712314161749632557`"},
            {"6", \(-0.00013130059838550192`\)},
            {"8", "1.4845532182493013`*^-6"},
            {"10", \(-1.13778653577798039`*^-8\)},
            {"12", "6.30329954060986263`*^-11"}
            },
          ColumnAlignments->{Decimal}], ")"}], TraditionalForm]], "Output"],

Cell["\<\
We see that the power series does not converge nearly as fast at \
the Chebyshev expansion.  However, since it does not involve an integration, \
it runs faster than the Chebyshev version.  This is why people use formulas \
to obtained from various trig identities to generate the algebraic equations \
for all of the derivatives.

For complex problems you would need exponential convergence and the power \
series method would not be a valid approach.\
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Eigenvalue problem", "Subtitle",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[{
  "We will solve the ODE, with an explicit parameter,\[Lambda] , for \
homogeneous boundary conditions.\n\n",
  Cell[BoxData[
      RowBox[{
        RowBox[{
          RowBox[{
            RowBox[{
              SuperscriptBox["u", "\[DoublePrime]",
                MultilineFunction->None], "[", "y", "]"}], "+", 
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Cell[CellGroupData[{

Cell["Here is the series expansion of the answer", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Series[anseig /. {A \[Rule] 1}, {y, 0, 8}]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    FormBox[
      InterpretationBox[
        RowBox[{
        "1", "+", \(1\/2\ \((\(-\[Lambda]\) - 1)\)\ y\^2\), "+", 
          \(1\/24\ \((\[Lambda] + 1)\)\^2\ y\^4\), "-", 
          \(1\/720\ \((\[Lambda] + 1)\)\^3\ y\^6\), "+", 
          \(\(\((\[Lambda] + 1)\)\^4\ y\^8\)\/40320\), "+", 
          InterpretationBox[\(O(y\^9)\),
            SeriesData[ y, 0, {}, 0, 9, 1]]}],
        SeriesData[ y, 0, {1, 0, 
          Times[ 
            Rational[ 1, 2], 
            Plus[ -1, 
              Times[ -1, \[Lambda]]]], 0, 
          Times[ 
            Rational[ 1, 24], 
            Power[ 
              Plus[ 1, \[Lambda]], 2]], 0, 
          Times[ 
            Rational[ -1, 720], 
            Power[ 
              Plus[ 1, \[Lambda]], 3]], 0, 
          Times[ 
            Rational[ 1, 40320], 
            Power[ 
              Plus[ 1, \[Lambda]], 4]]}, 0, 9, 1]], TraditionalForm]], 
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Cell[CellGroupData[{

Cell["Numerical Solution to the eigenvalue problem", "Subsection"],

Cell[CellGroupData[{

Cell["Computation of the eigenvalues", "Subsubsection"],

Cell["Here is our equation again.  ", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(odeeig1 = \[PartialD]\_{y, 2}vv[y] + \((1 + \[Lambda])\)\ vv[y]\)], 
  "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    FormBox[
      RowBox[{\(\((\[Lambda] + 1)\)\ \(vv(y)\)\), "+", 
        RowBox[{
          SuperscriptBox["vv", "\[DoublePrime]",
            MultilineFunction->None], "(", "y", ")"}]}], TraditionalForm]], 
  "Output"]
}, Open  ]],

Cell["Here is the standard substitution.", "Text"],

Cell[CellGroupData[{

Cell["\<\
t1 = odeeig1 /. 
   {vv[y] -> a[i]*ChebyshevT[i, y], 
    D[vv[y], {y, a1_}] -> D[a[i]*ChebyshevT[i, y], {y, a1}]}\
\>", "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\((\[Lambda] + 1)\)\ \(a(i)\)\ \(\(T\_i\)(y)\) + 
      \(i\ \(a(i)\)\ 
          \((i\ \(\(T\_i\)(y)\) - y\ \(\(U\_\(i - 1\)\)(y)\))\)\)\/\(y\^2 - 1
          \)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(t2 = ExpandAll[t1]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\(\(a(i)\)\ \(\(T\_i\)(y)\)\ i\^2\)\/\(y\^2 - 1\) - 
      \(y\ \(a(i)\)\ \(\(U\_\(i - 1\)\)(y)\)\ i\)\/\(y\^2 - 1\) + 
      \[Lambda]\ \(a(i)\)\ \(\(T\_i\)(y)\) + \(a(i)\)\ \(\(T\_i\)(y)\)\)], 
  "Output"]
}, Open  ]],

Cell["\<\
Now get a finite size sum to solve.  Note that we will need only \
even terms because the solution is clearly even.\
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(\(n = 12; \)\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    RowBox[{
      RowBox[{"t3", "=", 
        RowBox[{
          UnderoverscriptBox["\[Sum]", GridBox[{
                {\(i = 0\)},
                {\(\[CapitalDelta]\[MediumSpace]i = 2\)}
                }], "n"], "t2"}]}], ";"}]], "Input",
  AspectRatioFixed->True],

Cell["This might go quicker if we simplify it.", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(t4 = Simplify[t3]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`2048\ \[Lambda]\ \(a(12)\)\ y\^12 + 2048\ \(a(12)\)\ y\^12 + 
      512\ \[Lambda]\ \(a(10)\)\ y\^10 + 512\ \(a(10)\)\ y\^10 - 
      6144\ \[Lambda]\ \(a(12)\)\ y\^10 + 264192\ \(a(12)\)\ y\^10 + 
      128\ \[Lambda]\ \(a(8)\)\ y\^8 + 128\ \(a(8)\)\ y\^8 - 
      1280\ \[Lambda]\ \(a(10)\)\ y\^8 + 44800\ \(a(10)\)\ y\^8 + 
      6912\ \[Lambda]\ \(a(12)\)\ y\^8 - 546048\ \(a(12)\)\ y\^8 + 
      32\ \[Lambda]\ \(a(6)\)\ y\^6 + 32\ \(a(6)\)\ y\^6 - 
      256\ \[Lambda]\ \(a(8)\)\ y\^6 + 6912\ \(a(8)\)\ y\^6 + 
      1120\ \[Lambda]\ \(a(10)\)\ y\^6 - 70560\ \(a(10)\)\ y\^6 - 
      3584\ \[Lambda]\ \(a(12)\)\ y\^6 + 383488\ \(a(12)\)\ y\^6 + 
      8\ \[Lambda]\ \(a(4)\)\ y\^4 + 8\ \(a(4)\)\ y\^4 - 
      48\ \[Lambda]\ \(a(6)\)\ y\^4 + 912\ \(a(6)\)\ y\^4 + 
      160\ \[Lambda]\ \(a(8)\)\ y\^4 - 7520\ \(a(8)\)\ y\^4 - 
      400\ \[Lambda]\ \(a(10)\)\ y\^4 + 33200\ \(a(10)\)\ y\^4 + 
      840\ \[Lambda]\ \(a(12)\)\ y\^4 - 106680\ \(a(12)\)\ y\^4 - 
      8\ \[Lambda]\ \(a(4)\)\ y\^2 + 88\ \(a(4)\)\ y\^2 + 
      18\ \[Lambda]\ \(a(6)\)\ y\^2 - 558\ \(a(6)\)\ y\^2 - 
      32\ \[Lambda]\ \(a(8)\)\ y\^2 + 1888\ \(a(8)\)\ y\^2 + 
      50\ \[Lambda]\ \(a(10)\)\ y\^2 - 4750\ \(a(10)\)\ y\^2 - 
      72\ \[Lambda]\ \(a(12)\)\ y\^2 + 10008\ \(a(12)\)\ y\^2 + 
      \((\[Lambda] + 1)\)\ \(a(0)\) + 
      \((2\ \[Lambda]\ y\^2 + 2\ y\^2 - \[Lambda] + 3)\)\ \(a(2)\) + 
      \[Lambda]\ \(a(4)\) - 15\ \(a(4)\) - \[Lambda]\ \(a(6)\) + 
      35\ \(a(6)\) + \[Lambda]\ \(a(8)\) - 63\ \(a(8)\) - 
      \[Lambda]\ \(a(10)\) + 99\ \(a(10)\) + \[Lambda]\ \(a(12)\) - 
      143\ \(a(12)\)\)], "Output"]
}, Open  ]],

Cell["\<\
We need to find equations for the a[i]'s.  We can use the \
orthonality property of the polynomials to pick out equations for each a[i].
   
Set up integrals\
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(t5 = 
      \(t4\ 1\/\@\(1 - y\^2\)\ ChebyshevT[i, y]\ 2\)\/\(\[Pi]\ c[i]\)\)], 
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Cell[BoxData[
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    \`\(1\/\(\[Pi]\ \@\(1 - y\^2\)\ \(c(i)\)\)\((
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            512\ \[Lambda]\ \(a(10)\)\ y\^10 + 512\ \(a(10)\)\ y\^10 - 
            6144\ \[Lambda]\ \(a(12)\)\ y\^10 + 264192\ \(a(12)\)\ y\^10 + 
            128\ \[Lambda]\ \(a(8)\)\ y\^8 + 128\ \(a(8)\)\ y\^8 - 
            1280\ \[Lambda]\ \(a(10)\)\ y\^8 + 44800\ \(a(10)\)\ y\^8 + 
            6912\ \[Lambda]\ \(a(12)\)\ y\^8 - 546048\ \(a(12)\)\ y\^8 + 
            32\ \[Lambda]\ \(a(6)\)\ y\^6 + 32\ \(a(6)\)\ y\^6 - 
            256\ \[Lambda]\ \(a(8)\)\ y\^6 + 6912\ \(a(8)\)\ y\^6 + 
            1120\ \[Lambda]\ \(a(10)\)\ y\^6 - 70560\ \(a(10)\)\ y\^6 - 
            3584\ \[Lambda]\ \(a(12)\)\ y\^6 + 383488\ \(a(12)\)\ y\^6 + 
            8\ \[Lambda]\ \(a(4)\)\ y\^4 + 8\ \(a(4)\)\ y\^4 - 
            48\ \[Lambda]\ \(a(6)\)\ y\^4 + 912\ \(a(6)\)\ y\^4 + 
            160\ \[Lambda]\ \(a(8)\)\ y\^4 - 7520\ \(a(8)\)\ y\^4 - 
            400\ \[Lambda]\ \(a(10)\)\ y\^4 + 33200\ \(a(10)\)\ y\^4 + 
            840\ \[Lambda]\ \(a(12)\)\ y\^4 - 106680\ \(a(12)\)\ y\^4 - 
            8\ \[Lambda]\ \(a(4)\)\ y\^2 + 88\ \(a(4)\)\ y\^2 + 
            18\ \[Lambda]\ \(a(6)\)\ y\^2 - 558\ \(a(6)\)\ y\^2 - 
            32\ \[Lambda]\ \(a(8)\)\ y\^2 + 1888\ \(a(8)\)\ y\^2 + 
            50\ \[Lambda]\ \(a(10)\)\ y\^2 - 4750\ \(a(10)\)\ y\^2 - 
            72\ \[Lambda]\ \(a(12)\)\ y\^2 + 10008\ \(a(12)\)\ y\^2 + 
            \((\[Lambda] + 1)\)\ \(a(0)\) + 
            \((2\ \[Lambda]\ y\^2 + 2\ y\^2 - \[Lambda] + 3)\)\ \(a(2)\) + 
            \[Lambda]\ \(a(4)\) - 15\ \(a(4)\) - \[Lambda]\ \(a(6)\) + 
            35\ \(a(6)\) + \[Lambda]\ \(a(8)\) - 63\ \(a(8)\) - 
            \[Lambda]\ \(a(10)\) + 99\ \(a(10)\) + \[Lambda]\ \(a(12)\) - 
            143\ \(a(12)\))\)\ \(\(T\_i\)(y)\))\)\)\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(t6 = Table[t5, {i, 0, n, 2}]; \)\)], "Input",
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Cell[BoxData[
    \(\(t7 = \((t6 /. c[0] \[Rule] 2)\) /. Table[c[i] \[Rule] 1, {i, 2, n}]; 
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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                400\ \[Lambda]\ \(a(10)\) + 33200\ \(a(10)\) + 
                840\ \[Lambda]\ \(a(12)\) - 106680\ \(a(12)\)\), 
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                256\ \[Lambda]\ \(a(8)\) + 6912\ \(a(8)\) + 
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                3584\ \[Lambda]\ \(a(12)\) + 383488\ \(a(12)\)\), 
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                2\ \[Lambda]\ \(a(2)\) - 6\ \(a(2)\) - 
                2\ \[Lambda]\ \(a(4)\) + 30\ \(a(4)\) + 
                2\ \[Lambda]\ \(a(6)\) - 70\ \(a(6)\) - 
                2\ \[Lambda]\ \(a(8)\) + 126\ \(a(8)\) + 
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                2\ \[Lambda]\ \(a(12)\) + 286\ \(a(12)\)\), 
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                8\ \[Lambda]\ \(a(2)\) + 8\ \(a(2)\) + 
                20\ \[Lambda]\ \(a(4)\) - 236\ \(a(4)\) - 
                40\ \[Lambda]\ \(a(6)\) + 1256\ \(a(6)\) + 
                68\ \[Lambda]\ \(a(8)\) - 4028\ \(a(8)\) - 
                104\ \[Lambda]\ \(a(10)\) + 9896\ \(a(10)\) + 
                148\ \[Lambda]\ \(a(12)\) - 20588\ \(a(12)\)\), 
              \(8\ \[Lambda]\ \(a(2)\) + 8\ \(a(2)\) - 
                48\ \[Lambda]\ \(a(4)\) + 336\ \(a(4)\) + 
                168\ \[Lambda]\ \(a(6)\) - 4056\ \(a(6)\) - 
                448\ \[Lambda]\ \(a(8)\) + 22592\ \(a(8)\) + 
                1000\ \[Lambda]\ \(a(10)\) - 85400\ \(a(10)\) - 
                1968\ \[Lambda]\ \(a(12)\) + 253392\ \(a(12)\)\), 
              \(32\ \[Lambda]\ \(a(4)\) + 32\ \(a(4)\) - 
                256\ \[Lambda]\ \(a(6)\) + 3584\ \(a(6)\) + 
                1152\ \[Lambda]\ \(a(8)\) - 43904\ \(a(8)\) - 
                3840\ \[Lambda]\ \(a(10)\) + 273920\ \(a(10)\) + 
                10528\ \[Lambda]\ \(a(12)\) - 1193696\ \(a(12)\)\), 
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                1280\ \[Lambda]\ \(a(8)\) + 27392\ \(a(8)\) + 
                7040\ \[Lambda]\ \(a(10)\) - 371840\ \(a(10)\) - 
                28160\ \[Lambda]\ \(a(12)\) + 2626048\ \(a(12)\)\), 
              \(512\ \[Lambda]\ \(a(8)\) + 512\ \(a(8)\) - 
                6144\ \[Lambda]\ \(a(10)\) + 178176\ \(a(10)\) + 
                39936\ \[Lambda]\ \(a(12)\) - 2712576\ \(a(12)\)\), 
              \(2048\ \[Lambda]\ \(a(10)\) + 2048\ \(a(10)\) - 
                28672\ \[Lambda]\ \(a(12)\) + 1052672\ \(a(12)\)\), 
              \(8192\ \[Lambda]\ \(a(12)\) + 8192\ \(a(12)\)\), "0", "0", 
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                2\ \[Lambda]\ \(a(2)\) + 6\ \(a(2)\) + 
                2\ \[Lambda]\ \(a(4)\) - 30\ \(a(4)\) - 
                2\ \[Lambda]\ \(a(6)\) + 70\ \(a(6)\) + 
                2\ \[Lambda]\ \(a(8)\) - 126\ \(a(8)\) - 
                2\ \[Lambda]\ \(a(10)\) + 198\ \(a(10)\) + 
                2\ \[Lambda]\ \(a(12)\) - 286\ \(a(12)\)\), 
              \(\(-16\)\ \[Lambda]\ \(a(0)\) - 16\ \(a(0)\) + 
                20\ \[Lambda]\ \(a(2)\) - 44\ \(a(2)\) - 
                32\ \[Lambda]\ \(a(4)\) + 416\ \(a(4)\) + 
                52\ \[Lambda]\ \(a(6)\) - 1676\ \(a(6)\) - 
                80\ \[Lambda]\ \(a(8)\) + 4784\ \(a(8)\) + 
                116\ \[Lambda]\ \(a(10)\) - 11084\ \(a(10)\) - 
                160\ \[Lambda]\ \(a(12)\) + 22304\ \(a(12)\)\), 
              \(16\ \[Lambda]\ \(a(0)\) + 16\ \(a(0)\) - 
                48\ \[Lambda]\ \(a(2)\) + 16\ \(a(2)\) + 
                160\ \[Lambda]\ \(a(4)\) - 1632\ \(a(4)\) - 
                400\ \[Lambda]\ \(a(6)\) + 11312\ \(a(6)\) + 
                848\ \[Lambda]\ \(a(8)\) - 46256\ \(a(8)\) - 
                1616\ \[Lambda]\ \(a(10)\) + 143984\ \(a(10)\) + 
                2848\ \[Lambda]\ \(a(12)\) - 375776\ \(a(12)\)\), 
              \(32\ \[Lambda]\ \(a(2)\) + 32\ \(a(2)\) - 
                256\ \[Lambda]\ \(a(4)\) + 1280\ \(a(4)\) + 
                1120\ \[Lambda]\ \(a(6)\) - 23456\ \(a(6)\) - 
                3584\ \[Lambda]\ \(a(8)\) + 164352\ \(a(8)\) + 
                9440\ \[Lambda]\ \(a(10)\) - 748320\ \(a(10)\) - 
                21760\ \[Lambda]\ \(a(12)\) + 2633984\ \(a(12)\)\), 
              \(128\ \[Lambda]\ \(a(4)\) + 128\ \(a(4)\) - 
                1280\ \[Lambda]\ \(a(6)\) + 14080\ \(a(6)\) + 
                6912\ \[Lambda]\ \(a(8)\) - 230656\ \(a(8)\) - 
                26880\ \[Lambda]\ \(a(10)\) + 1749760\ \(a(10)\) + 
                84608\ \[Lambda]\ \(a(12)\) - 8934784\ \(a(12)\)\), 
              \(512\ \[Lambda]\ \(a(6)\) + 512\ \(a(6)\) - 
                6144\ \[Lambda]\ \(a(8)\) + 108544\ \(a(8)\) + 
                39424\ \[Lambda]\ \(a(10)\) - 1844736\ \(a(10)\) - 
                180224\ \[Lambda]\ \(a(12)\) + 15400960\ \(a(12)\)\), 
              \(2048\ \[Lambda]\ \(a(8)\) + 2048\ \(a(8)\) - 
                28672\ \[Lambda]\ \(a(10)\) + 708608\ \(a(10)\) + 
                212992\ \[Lambda]\ \(a(12)\) - 12959744\ \(a(12)\)\), 
              \(8192\ \[Lambda]\ \(a(10)\) + 8192\ \(a(10)\) - 
                131072\ \[Lambda]\ \(a(12)\) + 4194304\ \(a(12)\)\), 
              \(32768\ \[Lambda]\ \(a(12)\) + 32768\ \(a(12)\)\), "0", "0", 
              "0", "0"},
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                2\ \[Lambda]\ \(a(2)\) - 6\ \(a(2)\) - 
                2\ \[Lambda]\ \(a(4)\) + 30\ \(a(4)\) + 
                2\ \[Lambda]\ \(a(6)\) - 70\ \(a(6)\) - 
                2\ \[Lambda]\ \(a(8)\) + 126\ \(a(8)\) + 
                2\ \[Lambda]\ \(a(10)\) - 198\ \(a(10)\) - 
                2\ \[Lambda]\ \(a(12)\) + 286\ \(a(12)\)\), 
              \(36\ \[Lambda]\ \(a(0)\) + 36\ \(a(0)\) - 
                40\ \[Lambda]\ \(a(2)\) + 104\ \(a(2)\) + 
                52\ \[Lambda]\ \(a(4)\) - 716\ \(a(4)\) - 
                72\ \[Lambda]\ \(a(6)\) + 2376\ \(a(6)\) + 
                100\ \[Lambda]\ \(a(8)\) - 6044\ \(a(8)\) - 
                136\ \[Lambda]\ \(a(10)\) + 13064\ \(a(10)\) + 
                180\ \[Lambda]\ \(a(12)\) - 25164\ \(a(12)\)\), 
              \(\(-96\)\ \[Lambda]\ \(a(0)\) - 96\ \(a(0)\) + 
                168\ \[Lambda]\ \(a(2)\) - 216\ \(a(2)\) - 
                400\ \[Lambda]\ \(a(4)\) + 4592\ \(a(4)\) + 
                840\ \[Lambda]\ \(a(6)\) - 25272\ \(a(6)\) - 
                1568\ \[Lambda]\ \(a(8)\) + 89056\ \(a(8)\) + 
                2696\ \[Lambda]\ \(a(10)\) - 246904\ \(a(10)\) - 
                4368\ \[Lambda]\ \(a(12)\) + 587376\ \(a(12)\)\), 
              \(64\ \[Lambda]\ \(a(0)\) + 64\ \(a(0)\) - 
                256\ \[Lambda]\ \(a(2)\) + 1120\ \[Lambda]\ \(a(4)\) - 
                9120\ \(a(4)\) - 3584\ \[Lambda]\ \(a(6)\) + 
                88576\ \(a(6)\) + 9408\ \[Lambda]\ \(a(8)\) - 
                469824\ \(a(8)\) - 21504\ \[Lambda]\ \(a(10)\) + 
                1798656\ \(a(10)\) + 44384\ \[Lambda]\ \(a(12)\) - 
                5577376\ \(a(12)\)\), 
              \(128\ \[Lambda]\ \(a(2)\) + 128\ \(a(2)\) - 
                1280\ \[Lambda]\ \(a(4)\) + 4864\ \(a(4)\) + 
                6912\ \[Lambda]\ \(a(6)\) - 122112\ \(a(6)\) - 
                26880\ \[Lambda]\ \(a(8)\) + 1091328\ \(a(8)\) + 
                84480\ \[Lambda]\ \(a(10)\) - 6120960\ \(a(10)\) - 
                228096\ \[Lambda]\ \(a(12)\) + 25779456\ \(a(12)\)\), 
              \(512\ \[Lambda]\ \(a(4)\) + 512\ \(a(4)\) - 
                6144\ \[Lambda]\ \(a(6)\) + 55296\ \(a(6)\) + 
                39424\ \[Lambda]\ \(a(8)\) - 1140224\ \(a(8)\) - 
                180224\ \[Lambda]\ \(a(10)\) + 10510336\ \(a(10)\) + 
                658944\ \[Lambda]\ \(a(12)\) - 63828480\ \(a(12)\)\), 
              \(2048\ \[Lambda]\ \(a(6)\) + 2048\ \(a(6)\) - 
                28672\ \[Lambda]\ \(a(8)\) + 430080\ \(a(8)\) + 
                212992\ \[Lambda]\ \(a(10)\) - 8798208\ \(a(10)\) - 
                1118208\ \[Lambda]\ \(a(12)\) + 86470656\ \(a(12)\)\), 
              \(8192\ \[Lambda]\ \(a(8)\) + 8192\ \(a(8)\) - 
                131072\ \[Lambda]\ \(a(10)\) + 2818048\ \(a(10)\) + 
                1105920\ \[Lambda]\ \(a(12)\) - 60235776\ \(a(12)\)\), 
              \(32768\ \[Lambda]\ \(a(10)\) + 32768\ \(a(10)\) - 
                589824\ \[Lambda]\ \(a(12)\) + 16711680\ \(a(12)\)\), 
              \(131072\ \[Lambda]\ \(a(12)\) + 131072\ \(a(12)\)\), "0", "0", 
              "0"},
            {
              \(2\ \[Lambda]\ \(a(0)\) + 2\ \(a(0)\) - 
                2\ \[Lambda]\ \(a(2)\) + 6\ \(a(2)\) + 
                2\ \[Lambda]\ \(a(4)\) - 30\ \(a(4)\) - 
                2\ \[Lambda]\ \(a(6)\) + 70\ \(a(6)\) + 
                2\ \[Lambda]\ \(a(8)\) - 126\ \(a(8)\) - 
                2\ \[Lambda]\ \(a(10)\) + 198\ \(a(10)\) + 
                2\ \[Lambda]\ \(a(12)\) - 286\ \(a(12)\)\), 
              \(\(-64\)\ \[Lambda]\ \(a(0)\) - 64\ \(a(0)\) + 
                68\ \[Lambda]\ \(a(2)\) - 188\ \(a(2)\) - 
                80\ \[Lambda]\ \(a(4)\) + 1136\ \(a(4)\) + 
                100\ \[Lambda]\ \(a(6)\) - 3356\ \(a(6)\) - 
                128\ \[Lambda]\ \(a(8)\) + 7808\ \(a(8)\) + 
                164\ \[Lambda]\ \(a(10)\) - 15836\ \(a(10)\) - 
                208\ \[Lambda]\ \(a(12)\) + 29168\ \(a(12)\)\), 
              \(320\ \[Lambda]\ \(a(0)\) + 320\ \(a(0)\) - 
                448\ \[Lambda]\ \(a(2)\) + 832\ \(a(2)\) + 
                848\ \[Lambda]\ \(a(4)\) - 10416\ \(a(4)\) - 
                1568\ \[Lambda]\ \(a(6)\) + 48736\ \(a(6)\) + 
                2688\ \[Lambda]\ \(a(8)\) - 156032\ \(a(8)\) - 
                4320\ \[Lambda]\ \(a(10)\) + 402080\ \(a(10)\) + 
                6608\ \[Lambda]\ \(a(12)\) - 899632\ \(a(12)\)\), 
              \(\(-512\)\ \[Lambda]\ \(a(0)\) - 512\ \(a(0)\) + 
                1152\ \[Lambda]\ \(a(2)\) - 896\ \(a(2)\) - 
                3584\ \[Lambda]\ \(a(4)\) + 35328\ \(a(4)\) + 
                9408\ \[Lambda]\ \(a(6)\) - 254784\ \(a(6)\) - 
                21504\ \[Lambda]\ \(a(8)\) + 1131520\ \(a(8)\) + 
                44352\ \[Lambda]\ \(a(10)\) - 3836608\ \(a(10)\) - 
                84480\ \[Lambda]\ \(a(12)\) + 10870272\ \(a(12)\)\), 
              \(256\ \[Lambda]\ \(a(0)\) + 256\ \(a(0)\) - 
                1280\ \[Lambda]\ \(a(2)\) - 256\ \(a(2)\) + 
                6912\ \[Lambda]\ \(a(4)\) - 46336\ \(a(4)\) - 
                26880\ \[Lambda]\ \(a(6)\) + 584448\ \(a(6)\) + 
                84480\ \[Lambda]\ \(a(8)\) - 3831296\ \(a(8)\) - 
                228096\ \[Lambda]\ \(a(10)\) + 17686784\ \(a(10)\) + 
                549120\ \[Lambda]\ \(a(12)\) - 64933632\ \(a(12)\)\), 
              \(512\ \[Lambda]\ \(a(2)\) + 512\ \(a(2)\) - 
                6144\ \[Lambda]\ \(a(4)\) + 18432\ \(a(4)\) + 
                39424\ \[Lambda]\ \(a(6)\) - 599552\ \(a(6)\) - 
                180224\ \[Lambda]\ \(a(8)\) + 6537216\ \(a(8)\) + 
                658944\ \[Lambda]\ \(a(10)\) - 43659776\ \(a(10)\) - 
                2050048\ \[Lambda]\ \(a(12)\) + 215373824\ \(a(12)\)\), 
              \(2048\ \[Lambda]\ \(a(4)\) + 2048\ \(a(4)\) - 
                28672\ \[Lambda]\ \(a(6)\) + 217088\ \(a(6)\) + 
                212992\ \[Lambda]\ \(a(8)\) - 5423104\ \(a(8)\) - 
                1118208\ \[Lambda]\ \(a(10)\) + 58929152\ \(a(10)\) + 
                4659200\ \[Lambda]\ \(a(12)\) - 415295488\ \(a(12)\)\), 
              \(8192\ \[Lambda]\ \(a(6)\) + 8192\ \(a(6)\) - 
                131072\ \[Lambda]\ \(a(8)\) + 1703936\ \(a(8)\) + 
                1105920\ \[Lambda]\ \(a(10)\) - 40837120\ \(a(10)\) - 
                6553600\ \[Lambda]\ \(a(12)\) + 462159872\ \(a(12)\)\), 
              \(32768\ \[Lambda]\ \(a(8)\) + 32768\ \(a(8)\) - 
                589824\ \[Lambda]\ \(a(10)\) + 11206656\ \(a(10)\) + 
                5570560\ \[Lambda]\ \(a(12)\) - 274399232\ \(a(12)\)\), 
              \(131072\ \[Lambda]\ \(a(10)\) + 131072\ \(a(10)\) - 
                2621440\ \[Lambda]\ \(a(12)\) + 66584576\ \(a(12)\)\), 
              \(524288\ \[Lambda]\ \(a(12)\) + 524288\ \(a(12)\)\), "0", 
              "0"},
            {
              \(\(-2\)\ \[Lambda]\ \(a(0)\) - 2\ \(a(0)\) + 
                2\ \[Lambda]\ \(a(2)\) - 6\ \(a(2)\) - 
                2\ \[Lambda]\ \(a(4)\) + 30\ \(a(4)\) + 
                2\ \[Lambda]\ \(a(6)\) - 70\ \(a(6)\) - 
                2\ \[Lambda]\ \(a(8)\) + 126\ \(a(8)\) + 
                2\ \[Lambda]\ \(a(10)\) - 198\ \(a(10)\) - 
                2\ \[Lambda]\ \(a(12)\) + 286\ \(a(12)\)\), 
              \(100\ \[Lambda]\ \(a(0)\) + 100\ \(a(0)\) - 
                104\ \[Lambda]\ \(a(2)\) + 296\ \(a(2)\) + 
                116\ \[Lambda]\ \(a(4)\) - 1676\ \(a(4)\) - 
                136\ \[Lambda]\ \(a(6)\) + 4616\ \(a(6)\) + 
                164\ \[Lambda]\ \(a(8)\) - 10076\ \(a(8)\) - 
                200\ \[Lambda]\ \(a(10)\) + 19400\ \(a(10)\) + 
                244\ \[Lambda]\ \(a(12)\) - 34316\ \(a(12)\)\), 
              \(\(-800\)\ \[Lambda]\ \(a(0)\) - 800\ \(a(0)\) + 
                1000\ \[Lambda]\ \(a(2)\) - 2200\ \(a(2)\) - 
                1616\ \[Lambda]\ \(a(4)\) + 20784\ \(a(4)\) + 
                2696\ \[Lambda]\ \(a(6)\) - 85624\ \(a(6)\) - 
                4320\ \[Lambda]\ \(a(8)\) + 254240\ \(a(8)\) + 
                6600\ \[Lambda]\ \(a(10)\) - 620600\ \(a(10)\) - 
                9680\ \[Lambda]\ \(a(12)\) + 1328560\ \(a(12)\)\), 
              \(2240\ \[Lambda]\ \(a(0)\) + 2240\ \(a(0)\) - 
                3840\ \[Lambda]\ \(a(2)\) + 5120\ \(a(2)\) + 
                9440\ \[Lambda]\ \(a(4)\) - 103200\ \(a(4)\) - 
                21504\ \[Lambda]\ \(a(6)\) + 615936\ \(a(6)\) + 
                44352\ \[Lambda]\ \(a(8)\) - 2417344\ \(a(8)\) - 
                84480\ \[Lambda]\ \(a(10)\) + 7482880\ \(a(10)\) + 
                151008\ \[Lambda]\ \(a(12)\) - 19761696\ \(a(12)\)\), 
              \(\(-2560\)\ \[Lambda]\ \(a(0)\) - 2560\ \(a(0)\) + 
                7040\ \[Lambda]\ \(a(2)\) - 3200\ \(a(2)\) - 
                26880\ \[Lambda]\ \(a(4)\) + 229120\ \(a(4)\) + 
                84480\ \[Lambda]\ \(a(6)\) - 2065920\ \(a(6)\) - 
                228096\ \[Lambda]\ \(a(8)\) + 11097344\ \(a(8)\) + 
                549120\ \[Lambda]\ \(a(10)\) - 44599040\ \(a(10)\) - 
                1208064\ \[Lambda]\ \(a(12)\) + 147568896\ \(a(12)\)\), 
              \(1024\ \[Lambda]\ \(a(0)\) + 1024\ \(a(0)\) - 
                6144\ \[Lambda]\ \(a(2)\) - 2048\ \(a(2)\) + 
                39424\ \[Lambda]\ \(a(4)\) - 222720\ \(a(4)\) - 
                180224\ \[Lambda]\ \(a(6)\) + 3481600\ \(a(6)\) + 
                658944\ \[Lambda]\ \(a(8)\) - 27259392\ \(a(8)\) - 
                2050048\ \[Lambda]\ \(a(10)\) + 147556352\ \(a(10)\) + 
                5637632\ \[Lambda]\ \(a(12)\) - 626653696\ \(a(12)\)\), 
              \(2048\ \[Lambda]\ \(a(2)\) + 2048\ \(a(2)\) - 
                28672\ \[Lambda]\ \(a(4)\) + 69632\ \(a(4)\) + 
                212992\ \[Lambda]\ \(a(6)\) - 2834432\ \(a(6)\) - 
                1118208\ \[Lambda]\ \(a(8)\) + 36564992\ \(a(8)\) + 
                4659200\ \[Lambda]\ \(a(10)\) - 283699200\ \(a(10)\) - 
                16400384\ \[Lambda]\ \(a(12)\) + 1605615616\ \(a(12)\)\), 
              \(8192\ \[Lambda]\ \(a(4)\) + 8192\ \(a(4)\) - 
                131072\ \[Lambda]\ \(a(6)\) + 851968\ \(a(6)\) + 
                1105920\ \[Lambda]\ \(a(8)\) - 25108480\ \(a(8)\) - 
                6553600\ \[Lambda]\ \(a(10)\) + 314572800\ \(a(10)\) + 
                30638080\ \[Lambda]\ \(a(12)\) - 2525265920\ \(a(12)\)\), 
              \(32768\ \[Lambda]\ \(a(6)\) + 32768\ \(a(6)\) - 
                589824\ \[Lambda]\ \(a(8)\) + 6750208\ \(a(8)\) + 
                5570560\ \[Lambda]\ \(a(10)\) - 185794560\ \(a(10)\) - 
                36765696\ \[Lambda]\ \(a(12)\) + 2380726272\ \(a(12)\)\), 
              \(131072\ \[Lambda]\ \(a(8)\) + 131072\ \(a(8)\) - 
                2621440\ \[Lambda]\ \(a(10)\) + 44564480\ \(a(10)\) + 
                27394048\ \[Lambda]\ \(a(12)\) - 1230897152\ \(a(12)\)\), 
              \(524288\ \[Lambda]\ \(a(10)\) + 524288\ \(a(10)\) - 
                11534336\ \[Lambda]\ \(a(12)\) + 265289728\ \(a(12)\)\), 
              \(2097152\ \[Lambda]\ \(a(12)\) + 2097152\ \(a(12)\)\), "0"},
            {\(2\ \[Lambda]\ \(a(0)\) + 2\ \(a(0)\) - 
                2\ \[Lambda]\ \(a(2)\) + 6\ \(a(2)\) + 
                2\ \[Lambda]\ \(a(4)\) - 30\ \(a(4)\) - 
                2\ \[Lambda]\ \(a(6)\) + 70\ \(a(6)\) + 
                2\ \[Lambda]\ \(a(8)\) - 126\ \(a(8)\) - 
                2\ \[Lambda]\ \(a(10)\) + 198\ \(a(10)\) + 
                2\ \[Lambda]\ \(a(12)\) - 286\ \(a(12)\)\), 
              \(\(-144\)\ \[Lambda]\ \(a(0)\) - 144\ \(a(0)\) + 
                148\ \[Lambda]\ \(a(2)\) - 428\ \(a(2)\) - 
                160\ \[Lambda]\ \(a(4)\) + 2336\ \(a(4)\) + 
                180\ \[Lambda]\ \(a(6)\) - 6156\ \(a(6)\) - 
                208\ \[Lambda]\ \(a(8)\) + 12848\ \(a(8)\) + 
                244\ \[Lambda]\ \(a(10)\) - 23756\ \(a(10)\) - 
                288\ \[Lambda]\ \(a(12)\) + 40608\ \(a(12)\)\), 
              \(1680\ \[Lambda]\ \(a(0)\) + 1680\ \(a(0)\) - 
                1968\ \[Lambda]\ \(a(2)\) + 4752\ \(a(2)\) + 
                2848\ \[Lambda]\ \(a(4)\) - 37856\ \(a(4)\) - 
                4368\ \[Lambda]\ \(a(6)\) + 140976\ \(a(6)\) + 
                6608\ \[Lambda]\ \(a(8)\) - 392752\ \(a(8)\) - 
                9680\ \[Lambda]\ \(a(10)\) + 916720\ \(a(10)\) + 
                13728\ \[Lambda]\ \(a(12)\) - 1894752\ \(a(12)\)\), 
              \(\(-7168\)\ \[Lambda]\ \(a(0)\) - 7168\ \(a(0)\) + 
                10528\ \[Lambda]\ \(a(2)\) - 18144\ \(a(2)\) - 
                21760\ \[Lambda]\ \(a(4)\) + 254208\ \(a(4)\) + 
                44384\ \[Lambda]\ \(a(6)\) - 1319584\ \(a(6)\) - 
                84480\ \[Lambda]\ \(a(8)\) + 4720128\ \(a(8)\) + 
                151008\ \[Lambda]\ \(a(10)\) - 13611552\ \(a(10)\) - 
                256256\ \[Lambda]\ \(a(12)\) + 33967360\ \(a(12)\)\), 
              \(13824\ \[Lambda]\ \(a(0)\) + 13824\ \(a(0)\) - 
                28160\ \[Lambda]\ \(a(2)\) + 27136\ \(a(2)\) + 
                84608\ \[Lambda]\ \(a(4)\) - 824704\ \(a(4)\) - 
                228096\ \[Lambda]\ \(a(6)\) + 6011136\ \(a(6)\) + 
                549120\ \[Lambda]\ \(a(8)\) - 28032768\ \(a(8)\) - 
                1208064\ \[Lambda]\ \(a(10)\) + 101442816\ \(a(10)\) + 
                2471040\ \[Lambda]\ \(a(12)\) - 309250944\ \(a(12)\)\), 
              \(\(-12288\)\ \[Lambda]\ \(a(0)\) - 12288\ \(a(0)\) + 
                39936\ \[Lambda]\ \(a(2)\) - 9216\ \(a(2)\) - 
                180224\ \[Lambda]\ \(a(4)\) + 1343488\ \(a(4)\) + 
                658944\ \[Lambda]\ \(a(6)\) - 14627328\ \(a(6)\) - 
                2050048\ \[Lambda]\ \(a(8)\) + 92370944\ \(a(8)\) + 
                5637632\ \[Lambda]\ \(a(10)\) - 429849088\ \(a(10)\) - 
                14057472\ \[Lambda]\ \(a(12)\) + 1628209152\ \(a(12)\)\), 
              \(4096\ \[Lambda]\ \(a(0)\) + 4096\ \(a(0)\) - 
                28672\ \[Lambda]\ \(a(2)\) - 12288\ \(a(2)\) + 
                212992\ \[Lambda]\ \(a(4)\) - 1032192\ \(a(4)\) - 
                1118208\ \[Lambda]\ \(a(6)\) + 19378176\ \(a(6)\) + 
                4659200\ \[Lambda]\ \(a(8)\) - 176744448\ \(a(8)\) - 
                16400384\ \[Lambda]\ \(a(10)\) + 1098694656\ \(a(10)\) + 
                50692096\ \[Lambda]\ \(a(12)\) - 5302550528\ \(a(12)\)\), 
              \(8192\ \[Lambda]\ \(a(2)\) + 8192\ \(a(2)\) - 
                131072\ \[Lambda]\ \(a(4)\) + 262144\ \(a(4)\) + 
                1105920\ \[Lambda]\ \(a(6)\) - 13049856\ \(a(6)\) - 
                6553600\ \[Lambda]\ \(a(8)\) + 194772992\ \(a(8)\) + 
                30638080\ \[Lambda]\ \(a(10)\) - 1723105280\ \(a(10)\) - 
                120324096\ \[Lambda]\ \(a(12)\) + 11010834432\ \(a(12)\)\), 
              \(32768\ \[Lambda]\ \(a(4)\) + 32768\ \(a(4)\) - 
                589824\ \[Lambda]\ \(a(6)\) + 3342336\ \(a(6)\) + 
                5570560\ \[Lambda]\ \(a(8)\) - 113967104\ \(a(8)\) - 
                36765696\ \[Lambda]\ \(a(10)\) + 1618673664\ \(a(10)\) + 
                190513152\ \[Lambda]\ \(a(12)\) - 14588116992\ \(a(12)\)\), 
              \(131072\ \[Lambda]\ \(a(6)\) + 131072\ \(a(6)\) - 
                2621440\ \[Lambda]\ \(a(8)\) + 26738688\ \(a(8)\) + 
                27394048\ \[Lambda]\ \(a(10)\) - 832438272\ \(a(10)\) - 
                199229440\ \[Lambda]\ \(a(12)\) + 11918114816\ \(a(12)\)\), 
              \(524288\ \[Lambda]\ \(a(8)\) + 524288\ \(a(8)\) - 
                11534336\ \[Lambda]\ \(a(10)\) + 177209344\ \(a(10)\) + 
                132120576\ \[Lambda]\ \(a(12)\) - 5454692352\ \(a(12)\)\), 
              \(2097152\ \[Lambda]\ \(a(10)\) + 2097152\ \(a(10)\) - 
                50331648\ \[Lambda]\ \(a(12)\) + 1056964608\ \(a(12)\)\), 
              \(8388608\ \[Lambda]\ \(a(12)\) + 8388608\ \(a(12)\)\)}
            }], ")"}], TraditionalForm]], "Output"]
}, Open  ]],

Cell["Here are the factors that we pulled out.  ", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(t10 = Table[y\^i\/\(\[Pi]\ \@\(1 - y\^2\)\), {i, 0, 2\ n, 2}]\)], 
  "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`{1\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^2\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^4\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^6\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^8\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^10\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^12\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^14\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^16\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^18\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^20\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^22\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^24\/\(\[Pi]\ \@\(1 - y\^2\)\)}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Now we just need to integrate each of them once.  If we had left them \
interspersed in the equations, we would have done about ",
  Cell[BoxData[
      \(TraditionalForm\`\((2  n)\)\^3\)]],
  " integrations!"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Timing[t11 = \[Integral]\_\(-1\)\%1 t10 \[DifferentialD]y]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`{2.63333333333333463`\ Second, {1, 1\/2, 3\/8, 5\/16, 35\/128, 63\/256, 
        231\/1024, 429\/2048, 6435\/32768, 12155\/65536, 46189\/262144, 
        88179\/524288, 676039\/4194304}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(t12 = Simplify[Expand[t9 . t11]]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`{\((\[Lambda] + 1)\)\ \(a(0)\) + 
        4\ \((a(2) + 8\ \(a(4)\) + 27\ \(a(6)\) + 64\ \(a(8)\) + 
              125\ \(a(10)\) + 216\ \(a(12)\))\), 
      \((\[Lambda] + 1)\)\ \(a(2)\) + 
        48\ \((a(4) + 4\ \(a(6)\) + 10\ \(a(8)\) + 20\ \(a(10)\) + 
              35\ \(a(12)\))\), 
      \((\[Lambda] + 1)\)\ \(a(4)\) + 
        24\ \((5\ \(a(6)\) + 16\ \(a(8)\) + 35\ \(a(10)\) + 64\ \(a(12)\))\), 
      \((\[Lambda] + 1)\)\ \(a(6)\) + 
        16\ \((14\ \(a(8)\) + 40\ \(a(10)\) + 81\ \(a(12)\))\), 
      \((\[Lambda] + 1)\)\ \(a(8)\) + 120\ \((3\ \(a(10)\) + 8\ \(a(12)\))\), 
      \((\[Lambda] + 1)\)\ \(a(10)\) + 528\ \(a(12)\), 
      \((\[Lambda] + 1)\)\ \(a(12)\)}\)], "Output"]
}, Open  ]],

Cell[TextData[
"Here is the eigenvalue problem from the ODE.  At this point, we have not \
enforced the boundary conditions so we cannot just solve this matrix for \
\[Lambda].  "], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(t13 = Transpose[Table[Coefficient[t12, a[i]], {i, 0, n, 2}]]\)], "Input",\

  AspectRatioFixed->True],

Cell[BoxData[
    FormBox[
      RowBox[{"(", GridBox[{
            {\(\[Lambda] + 1\), "4", "32", "108", "256", "500", "864"},
            {"0", \(\[Lambda] + 1\), "48", "192", "480", "960", "1680"},
            {"0", "0", \(\[Lambda] + 1\), "120", "384", "840", "1536"},
            {"0", "0", "0", \(\[Lambda] + 1\), "224", "640", "1296"},
            {"0", "0", "0", "0", \(\[Lambda] + 1\), "360", "960"},
            {"0", "0", "0", "0", "0", \(\[Lambda] + 1\), "528"},
            {"0", "0", "0", "0", "0", "0", \(\[Lambda] + 1\)}
            }], ")"}], TraditionalForm]], "Output"]
}, Open  ]],

Cell["Here is the homogeneous boundary condition at y=1.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(b1 = Table[1, {i, 0, n, 2}]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`{1, 1, 1, 1, 1, 1, 1}\)], "Output"]
}, Open  ]],

Cell["\<\
To enforce the boundary condition, we drop off the last equation \
(highest mode) and replace it with the boundary condition.  If course, this \
makes the eigenvalue problem singular.  \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(t14 = 
      Append[Table[
          t13\[LeftDoubleBracket]i\[RightDoubleBracket], {i, 1, n\/2}], b1]
        \)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    FormBox[
      RowBox[{"(", GridBox[{
            {\(\[Lambda] + 1\), "4", "32", "108", "256", "500", "864"},
            {"0", \(\[Lambda] + 1\), "48", "192", "480", "960", "1680"},
            {"0", "0", \(\[Lambda] + 1\), "120", "384", "840", "1536"},
            {"0", "0", "0", \(\[Lambda] + 1\), "224", "640", "1296"},
            {"0", "0", "0", "0", \(\[Lambda] + 1\), "360", "960"},
            {"0", "0", "0", "0", "0", \(\[Lambda] + 1\), "528"},
            {"1", "1", "1", "1", "1", "1", "1"}
            }], ")"}], TraditionalForm]], "Output"]
}, Open  ]],

Cell["\<\
Because eigenvalue problem is singular, we need to algebraically \
remove the last equation.  We will solve for the highest coefficient, a[n], \
and then use result this to eliminate a[n] everywhere in the array.

Here we just take the last equation (by taking the last row, multiplying by \
the a[i]'s) and then solve for a[n].  \
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(s1 = 
      Solve[t14\[LeftDoubleBracket]n\/2 + 1\[RightDoubleBracket] . 
            Table[a[i], {i, 0, n, 2}] == 0, a[n]]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`{{a(12) \[Rule] \(-\(a(0)\)\) - a(2) - a(4) - a(6) - a(8) - a(10)}}\)], 
  "Output"]
}, Open  ]],

Cell["\<\
Let's be inelegant.  Remake the set of equations and then do the \
elimination of a[n] in the rest of the system.  (You can redo this another \
way!)\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(t15 = 
      Table[t14\[LeftDoubleBracket]i\[RightDoubleBracket] . 
          Table[a[i], {i, 0, n, 2}], {i, 1, n\/2}]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`{\((\[Lambda] + 1)\)\ \(a(0)\) + 4\ \(a(2)\) + 32\ \(a(4)\) + 
        108\ \(a(6)\) + 256\ \(a(8)\) + 500\ \(a(10)\) + 864\ \(a(12)\), 
      \((\[Lambda] + 1)\)\ \(a(2)\) + 48\ \(a(4)\) + 192\ \(a(6)\) + 
        480\ \(a(8)\) + 960\ \(a(10)\) + 1680\ \(a(12)\), 
      \((\[Lambda] + 1)\)\ \(a(4)\) + 120\ \(a(6)\) + 384\ \(a(8)\) + 
        840\ \(a(10)\) + 1536\ \(a(12)\), 
      \((\[Lambda] + 1)\)\ \(a(6)\) + 224\ \(a(8)\) + 640\ \(a(10)\) + 
        1296\ \(a(12)\), 
      \((\[Lambda] + 1)\)\ \(a(8)\) + 360\ \(a(10)\) + 960\ \(a(12)\), 
      \((\[Lambda] + 1)\)\ \(a(10)\) + 528\ \(a(12)\)}\)], "Output"]
}, Open  ]],

Cell["\<\
Here is the substitution for a[n] which runs off of the page.  \
\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(t16 = Expand[t15 /. s1]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    FormBox[
      RowBox[{"(", GridBox[{
            {
              \(\[Lambda]\ \(a(0)\) - 863\ \(a(0)\) - 860\ \(a(2)\) - 
                832\ \(a(4)\) - 756\ \(a(6)\) - 608\ \(a(8)\) - 
                364\ \(a(10)\)\), 
              \(\(-1680\)\ \(a(0)\) + \[Lambda]\ \(a(2)\) - 1679\ \(a(2)\) - 
                1632\ \(a(4)\) - 1488\ \(a(6)\) - 1200\ \(a(8)\) - 
                720\ \(a(10)\)\), 
              \(\(-1536\)\ \(a(0)\) - 1536\ \(a(2)\) + \[Lambda]\ \(a(4)\) - 
                1535\ \(a(4)\) - 1416\ \(a(6)\) - 1152\ \(a(8)\) - 
                696\ \(a(10)\)\), 
              \(\(-1296\)\ \(a(0)\) - 1296\ \(a(2)\) - 1296\ \(a(4)\) + 
                \[Lambda]\ \(a(6)\) - 1295\ \(a(6)\) - 1072\ \(a(8)\) - 
                656\ \(a(10)\)\), 
              \(\(-960\)\ \(a(0)\) - 960\ \(a(2)\) - 960\ \(a(4)\) - 
                960\ \(a(6)\) + \[Lambda]\ \(a(8)\) - 959\ \(a(8)\) - 
                600\ \(a(10)\)\), 
              \(\(-528\)\ \(a(0)\) - 528\ \(a(2)\) - 528\ \(a(4)\) - 
                528\ \(a(6)\) - 528\ \(a(8)\) + \[Lambda]\ \(a(10)\) - 
                527\ \(a(10)\)\)}
            }], ")"}], TraditionalForm]], "Output"]
}, Open  ]],

Cell["\<\
Now remake the eigenvalue matrix, which is now of lower order, but \
which has all finite eigenvalues.\
\>", "Text",
  CellTags->"eigenvalues"],

Cell[CellGroupData[{

Cell[BoxData[
    \(t17 = 
      Transpose[
        Table[Table[
            Coefficient[t16\[LeftDoubleBracket]1, i\[RightDoubleBracket], 
              a[j]], {i, 1, n\/2}], {j, 0, n - 2, 2}]]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    FormBox[
      RowBox[{"(", GridBox[{
            {\(\[Lambda] - 863\), \(-860\), \(-832\), \(-756\), \(-608\), 
              \(-364\)},
            {\(-1680\), \(\[Lambda] - 1679\), \(-1632\), \(-1488\), 
              \(-1200\), \(-720\)},
            {\(-1536\), \(-1536\), \(\[Lambda] - 1535\), \(-1416\), 
              \(-1152\), \(-696\)},
            {\(-1296\), \(-1296\), \(-1296\), \(\[Lambda] - 1295\), 
              \(-1072\), \(-656\)},
            {\(-960\), \(-960\), \(-960\), \(-960\), \(\[Lambda] - 959\), 
              \(-600\)},
            {\(-528\), \(-528\), \(-528\), \(-528\), \(-528\), 
              \(\[Lambda] - 527\)}
            }], ")"}], TraditionalForm]], "Output"]
}, Open  ]],

Cell[TextData[{
  "Return to ",
  ButtonBox["conclusions",
    ButtonData:>"conclusions",
    ButtonStyle->"Hyperlink"]
}], "Text"],

Cell[TextData[{
  "Here we compute the eigenvalues.  Note the insertion of the - sign.   \n\n\
We expect them to decrease in accuracy.  Thus if we want ",
  StyleBox["n",
    FontSlant->"Italic"],
  " eigenvalues, we might need ",
  StyleBox["n",
    FontSlant->"Italic"],
  "+4 or more modes in the original expansion.\n\n",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " sorts these for us and does not seem to complain about computing them.  I \
would not try to get them algebraically however!  "
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(neigs = \ N[Eigenvalues[\(-t17\) /. \[Lambda] \[Rule] 0]]\ \)\)], 
  "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`{1.46740110027259103`, 21.2066082218780005`, 60.7158402643227645`, 
      126.82990386032098`, 359.86087363195609`, 6287.91937292124902`}\)], 
  "Output"]
}, Open  ]],

Cell["\<\
Here are the analytically determined eigenvalues from above.  Note \
how the accuracy decreases as they get bigger.  If you need them very \
accurately, you will need several more modes than the number of eigenvalues \
you need.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(eigsanlyt\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`{1.46740110027233949`, 21.2066099024510545`, 60.6850275068084865`, 
      119.902653913344646`, 198.859489122059507`, 297.555533132953087`, 
      415.990785946025365`}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "At this point, for many problems, ",
  StyleBox["you might be finished",
    FontVariations->{"Underline"->True}],
  ".  You have computed the eigenvalues."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Computation of the eigenfunctions", "Subsubsection"],

Cell["\<\
If you need the eigenfunctions, that correspond to each of the \
eigenvalues, you will have to solve the matrix, t17, which enforces the \
boundary condition, for the a[i]'s.  Of course, there cannot be a unique \
solution because this is an eigenvalue problem, matrix t17 is (it better be) \
singular.

(Note that the eigenfunctions are probably less accurate than the \
eigenvalues!!  Thus you will need even more modes to get the eigenfunctions \
correct.)

 Here we again remake the equations\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(t18 = 
      Table[t17\[LeftDoubleBracket]i\[RightDoubleBracket] . 
          Table[a[i], {i, 0, n - 2, 2}], {i, 1, n\/2 - 1}]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`{\((\[Lambda] - 863)\)\ \(a(0)\) - 860\ \(a(2)\) - 832\ \(a(4)\) - 
        756\ \(a(6)\) - 608\ \(a(8)\) - 364\ \(a(10)\), 
      \(-1680\)\ \(a(0)\) + \((\[Lambda] - 1679)\)\ \(a(2)\) - 
        1632\ \(a(4)\) - 1488\ \(a(6)\) - 1200\ \(a(8)\) - 720\ \(a(10)\), 
      \(-1536\)\ \(a(0)\) - 1536\ \(a(2)\) + 
        \((\[Lambda] - 1535)\)\ \(a(4)\) - 1416\ \(a(6)\) - 1152\ \(a(8)\) - 
        696\ \(a(10)\), 
      \(-1296\)\ \(a(0)\) - 1296\ \(a(2)\) - 1296\ \(a(4)\) + 
        \((\[Lambda] - 1295)\)\ \(a(6)\) - 1072\ \(a(8)\) - 656\ \(a(10)\), 
      \(-960\)\ \(a(0)\) - 960\ \(a(2)\) - 960\ \(a(4)\) - 960\ \(a(6)\) + 
        \((\[Lambda] - 959)\)\ \(a(8)\) - 600\ \(a(10)\)}\)], "Output"]
}, Open  ]],

Cell["Set them up to solve.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(t19 = LogicalExpand[t18 == 0]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\(-1680\)\ \(a(0)\) + \((\[Lambda] - 1679)\)\ \(a(2)\) - 
          1632\ \(a(4)\) - 1488\ \(a(6)\) - 1200\ \(a(8)\) - 
          720\ \(a(10)\) == 0 \[And] 
      \(-1536\)\ \(a(0)\) - 1536\ \(a(2)\) + 
          \((\[Lambda] - 1535)\)\ \(a(4)\) - 1416\ \(a(6)\) - 
          1152\ \(a(8)\) - 696\ \(a(10)\) == 0 \[And] 
      \(-1296\)\ \(a(0)\) - 1296\ \(a(2)\) - 1296\ \(a(4)\) + 
          \((\[Lambda] - 1295)\)\ \(a(6)\) - 1072\ \(a(8)\) - 
          656\ \(a(10)\) == 0 \[And] 
      \(-960\)\ \(a(0)\) - 960\ \(a(2)\) - 960\ \(a(4)\) - 960\ \(a(6)\) + 
          \((\[Lambda] - 959)\)\ \(a(8)\) - 600\ \(a(10)\) == 0 \[And] 
      \((\[Lambda] - 863)\)\ \(a(0)\) - 860\ \(a(2)\) - 832\ \(a(4)\) - 
          756\ \(a(6)\) - 608\ \(a(8)\) - 364\ \(a(10)\) == 0\)], "Output"]
}, Open  ]],

Cell["Solve them.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(t20 = Solve[t19, Table[a[i], {i, 2, n - 2, 2}]]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`{{a(2) \[Rule] 
          1\/860\ \((\[Lambda] - 863)\)\ \(a(0)\) - 
            \(756\ \((\[Lambda] + 1)\)\ 
                \((41\ \(a(0)\)\ \[Lambda]\^3 - 
                    22137\ \(a(0)\)\ \[Lambda]\^2 + 
                    525843\ \(a(0)\)\ \[Lambda] - 719179\ \(a(0)\))\)\)\/\(215
                \ \((91\ \[Lambda]\^4 + 17164\ \[Lambda]\^3 - 
                    813054\ \[Lambda]\^2 + 16201804\ \[Lambda] - 60382469)
                  \)\) - \(912\ 
                \((5\ \(a(0)\)\ \[Lambda]\^4 - 7372\ \(a(0)\)\ \[Lambda]\^3 + 
                    674814\ \(a(0)\)\ \[Lambda]\^2 - 
                    11807116\ \(a(0)\)\ \[Lambda] + 15895973\ \(a(0)\))
                  \)\)\/\(43\ 
                \((91\ \[Lambda]\^4 + 17164\ \[Lambda]\^3 - 
                    813054\ \[Lambda]\^2 + 16201804\ \[Lambda] - 60382469)
                  \)\) - \(1248\ 
                \((29\ \(a(0)\)\ \[Lambda]\^4 - 
                    2204\ \(a(0)\)\ \[Lambda]\^3 - 
                    63426\ \(a(0)\)\ \[Lambda]\^2 - 
                    120124\ \(a(0)\)\ \[Lambda] - 58931\ \(a(0)\))\)\)\/\(215
                \ \((91\ \[Lambda]\^4 + 17164\ \[Lambda]\^3 - 
                    813054\ \[Lambda]\^2 + 16201804\ \[Lambda] - 60382469)
                  \)\) + \((
                91\ \((\(-\(a(0)\)\)\ \[Lambda]\^5 + 
                      6331\ \(a(0)\)\ \[Lambda]\^4 - 
                      1938826\ \(a(0)\)\ \[Lambda]\^3 + 
                      129528566\ \(a(0)\)\ \[Lambda]\^2 - 
                      2065274501\ \(a(0)\)\ \[Lambda] + 2757773375\ \(a(0)\))
                    \))\)/\((
                860\ \((
                    91\ \[Lambda]\^4 + 17164\ \[Lambda]\^3 - 
                      813054\ \[Lambda]\^2 + 16201804\ \[Lambda] - 60382469)
                    \))\), a(4) \[Rule] 
          \(6\ \((29\ \(a(0)\)\ \[Lambda]\^4 - 2204\ \(a(0)\)\ \[Lambda]\^3 - 
                  63426\ \(a(0)\)\ \[Lambda]\^2 - 
                  120124\ \(a(0)\)\ \[Lambda] - 58931\ \(a(0)\))\)\)\/\(91\ 
                \[Lambda]\^4 + 17164\ \[Lambda]\^3 - 813054\ \[Lambda]\^2 + 
              16201804\ \[Lambda] - 60382469\), 
        a(6) \[Rule] 
          \(4\ \((\[Lambda] + 1)\)\ 
              \((41\ \(a(0)\)\ \[Lambda]\^3 - 22137\ \(a(0)\)\ \[Lambda]\^2 + 
                  525843\ \(a(0)\)\ \[Lambda] - 719179\ \(a(0)\))\)\)\/\(91\ 
                \[Lambda]\^4 + 17164\ \[Lambda]\^3 - 813054\ \[Lambda]\^2 + 
              16201804\ \[Lambda] - 60382469\), 
        a(8) \[Rule] 
          \(30\ \((
                5\ \(a(0)\)\ \[Lambda]\^4 - 7372\ \(a(0)\)\ \[Lambda]\^3 + 
                  674814\ \(a(0)\)\ \[Lambda]\^2 - 
                  11807116\ \(a(0)\)\ \[Lambda] + 15895973\ \(a(0)\))
                \)\)\/\(91\ \[Lambda]\^4 + 17164\ \[Lambda]\^3 - 
              813054\ \[Lambda]\^2 + 16201804\ \[Lambda] - 60382469\), 
        a(10) \[Rule] 
          \(-\(\((\(-\(a(0)\)\)\ \[Lambda]\^5 + 
                  6331\ \(a(0)\)\ \[Lambda]\^4 - 
                  1938826\ \(a(0)\)\ \[Lambda]\^3 + 
                  129528566\ \(a(0)\)\ \[Lambda]\^2 - 
                  2065274501\ \(a(0)\)\ \[Lambda] + 2757773375\ \(a(0)\))\)/
              \((4\ \((
                    91\ \[Lambda]\^4 + 17164\ \[Lambda]\^3 - 
                      813054\ \[Lambda]\^2 + 16201804\ \[Lambda] - 60382469)
                    \))\)\)\)}}\)], "Output"]
}, Open  ]],

Cell["\<\
Now we can compute the eigenfunctions for any of the eigenvalues.  \
Here is the basic expansion.  Note that I have left off the last term.\
\>", 
  "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{"expand2", "=", 
      RowBox[{
        UnderoverscriptBox["\[Sum]", GridBox[{
              {\(i = 0\)},
              {\(\[CapitalDelta]\[MediumSpace]i = 2\)}
              }], \(n - 2\)], \(a[i]\ ChebyshevT[i, y]\)}]}]], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`a(0) + \((2\ y\^2 - 1)\)\ \(a(2)\) + 
      \((8\ y\^4 - 8\ y\^2 + 1)\)\ \(a(4)\) + 
      \((32\ y\^6 - 48\ y\^4 + 18\ y\^2 - 1)\)\ \(a(6)\) + 
      \((128\ y\^8 - 256\ y\^6 + 160\ y\^4 - 32\ y\^2 + 1)\)\ \(a(8)\) + 
      \((512\ y\^10 - 1280\ y\^8 + 1120\ y\^6 - 400\ y\^4 + 50\ y\^2 - 1)\)\ 
        \(a(10)\)\)], "Output"]
}, Open  ]],

Cell[TextData[
"To get the eigenfunction, you just substitute the a[i]'s and the desired \
\[Lambda], like this.  Remember that the solution is not unique so you have \
an arbitrary scale factor, a[0], that you can pick.  For numerical problems \
you usually need to do some sort of normalization.  We can pick the a[0]'s so \
that each eigenfunction has a value of 1 at y=0.    "], "Text",
  CellTags->"eigenfunctions"],

Cell[TextData[{
  "Return to ",
  ButtonBox["conclusions",
    ButtonData:>"conclusions",
    ButtonStyle->"Hyperlink"]
}], "Text"],

Cell[CellGroupData[{

Cell[TextData["efunc1=((expand2/.t20)/.\[Lambda]->neigs[[1]])"], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`{\(-1.05805502525594175`\)\ \((2\ y\^2 - 1)\)\ \(a(0)\) + 
        0.0593051006701230853`\ \((8\ y\^4 - 8\ y\^2 + 1)\)\ \(a(0)\) - 
        0.00126418148219753989`\ \((32\ y\^6 - 48\ y\^4 + 18\ y\^2 - 1)\)\ 
          \(a(0)\) + 
        0.0000142041898105456665`\ 
          \((128\ y\^8 - 256\ y\^6 + 160\ y\^4 - 32\ y\^2 + 1)\)\ \(a(0)\) - 
        9.85824814438980467`*^-8\ 
          \((512\ y\^10 - 1280\ y\^8 + 1120\ y\^6 - 400\ y\^4 + 50\ y\^2 - 1)
            \)\ \(a(0)\) + a(0)}\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[
"Here are some plots of eigenfunctions for different values of \[Lambda].   \
"], "Subsubsection"],

Cell[CellGroupData[{

Cell["neigs", "Input",
  CellLabel->"In[28]:=",
  Evaluatable->False],

Cell[BoxData[
    \(TraditionalForm
    \`{1.46740110027259103`, 21.2066082218780005`, 60.7158402643227645`, 
      126.82990386032098`, 359.86087363195609`, 6287.91937292124902`}\)], 
  "Output",
  CellLabel->"Out[28]="]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData["\[Lambda] = 1.4674"], "Text"],

Cell[CellGroupData[{

Cell[TextData["efunc1=(expand2/.t20)/.\[Lambda]->neigs[[1]]"], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`{\(-1.05805502525594175`\)\ \((2\ y\^2 - 1)\)\ \(a(0)\) + 
        0.0593051006701230853`\ \((8\ y\^4 - 8\ y\^2 + 1)\)\ \(a(0)\) - 
        0.00126418148219753989`\ \((32\ y\^6 - 48\ y\^4 + 18\ y\^2 - 1)\)\ 
          \(a(0)\) + 
        0.0000142041898105456665`\ 
          \((128\ y\^8 - 256\ y\^6 + 160\ y\^4 - 32\ y\^2 + 1)\)\ \(a(0)\) - 
        9.85824814438980467`*^-8\ 
          \((512\ y\^10 - 1280\ y\^8 + 1120\ y\^6 - 400\ y\^4 + 50\ y\^2 - 1)
            \)\ \(a(0)\) + a(0)}\)], "Output"],

Cell["\<\
Here is where we normalize the eigenfunction to have a value of 1 \
at y=0.  \
\>", "Text",
  CellLabelAutoDelete->True],

Cell[CellGroupData[{

Cell["a01=a[0]/. Solve[((efunc1/.y->0)[[1]])==1,a[0]][[1]]", "Input"],

Cell[BoxData[
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}, Open  ]],

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eplot1=Plot[efunc1/.a[0]->a01,{y,-1,1},
PlotStyle->Dashing[{.02,.02}]];\
\>", "Input"],

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    \(TraditionalForm
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        3.3870189826171666`\ \((32\ y\^6 - 48\ y\^4 + 18\ y\^2 - 1)\)\ 
          \(a(0)\) + 
        2.04278158544454191`\ 
          \((128\ y\^8 - 256\ y\^6 + 160\ y\^4 - 32\ y\^2 + 1)\)\ \(a(0)\) - 
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Cell["\<\
Here is where we normalize the eigenfunction to have a value of 1 \
at y=0.  \
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
a03=a[0]/.Solve[((efunc3/.y->0)[[1]])==1,
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\>", "Input"],

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Cell["\<\
eplot3=Plot[efunc3/.a[0]->a03,{y,-1,1},
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\>", "Input"],

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MathPictureStart
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