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MathGroup Archive 1999

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Re: DSolve

  • To: mathgroup at smc.vnet.net
  • Subject: [mg18775] Re: DSolve
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Tue, 20 Jul 1999 01:33:31 -0400
  • Organization: University of Western Australia
  • References: <7mp2bh$l7j@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

"N. Shamsundar" wrote:

> Here is a pair of differential equations that Version 3 could not solve
> (actually, it was still trying after 15 minutes on a 400 MHz PII!)
>
> w:=Exp[-a x];
> DSolve[{u'[x]==-u[x]+v[x]+w  (1-a)+Cos[Pi x]-(1+Pi) Sin[Pi x],
>                 v''''[x]==u[x]+v[x]+(Pi^4-1) Sin[Pi x]-Cos[Pi x]-w, u[0]==2,
>                 v[0]==0,v''[0]==0,v[1]==0,v''[1]==0},
>                 {u[x],v[x]},x]
>
> These are linear equations with constant coefficients, and the exact solution
> is
>         u=w+cos(Pi x), v=sin(Pi x)
>
> I would appreciate someone running this calculation in Version 4.

The problem is not version specific.  If you try solving your pair of equations
without the boundary conditions you will get a rather long answer involving Root
objects.  If you eliminate u[x] you obtain a 5th order equation in v leading to
a fifth order auxilliary equation with non-trivial roots (and this is where
Mathematica gets stuck).

The following Notebook suggests a general approach for solving this type of
problem.

Notebook[{

Cell[CellGroupData[{
Cell["Modifiying Equal", "Section"],

Cell[TextData[{
  "We modify ",
  Cell[BoxData[
      FormBox[
        StyleBox["Equal",
          "Input"], TraditionalForm]]],
  " so that ",
  Cell[BoxData[
      FormBox[
        StyleBox["Listable",
          "Input"], TraditionalForm]]],
  " operations are automatically applied to both sides of any equality:"
}], "Text",
  CellTags->{"Equal", "Listable"}],

Cell[BoxData[
    \(TraditionalForm\`\(Unprotect[Equal];\)\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`listableQ(f_) :=
      MemberQ(Attributes(f), Listable)\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`Equal /: \
      lhs : f_Symbol?
            listableQ[___, \ _Equal, \ ___]\  := \n\ \ \ \ \ \ \ \ Thread[\
        Unevaluated[lhs], \ Equal\ ]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`\(Protect[Equal];\)\)], "Input"],

Cell["This allows direct manipulation of equations.", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Pair of Differential Equations", "Section"],

Cell[BoxData[
    \(TraditionalForm\`\(w[
          x_] = \[ExponentialE]\^\(\(-a\)\ x\);\)\)], "Input"],

Cell[BoxData[
    FormBox[
      RowBox[{
        RowBox[{\(\[ScriptCapitalE]\_1\), "=",
          RowBox[{
            RowBox[{
              SuperscriptBox["u", "\[Prime]",
                MultilineFunction->None], "(", "x", ")"}],
            "==", \(\((1 - a)\)\ w[x] +
              cos(\[Pi]\ x) - \((1 + \[Pi])\)\ \(sin(\[Pi]\ x)\) - u(x) +
              v(x)\)}]}], ";"}], TraditionalForm]], "Input"],

Cell[BoxData[
    FormBox[
      RowBox[{
        RowBox[{\(\[ScriptCapitalE]\_2\), "=",
          RowBox[{
            RowBox[{
              SuperscriptBox["v",
                TagBox[\((4)\),
                  Derivative],
                MultilineFunction->None], "(", "x", ")"}],
            "==", \(\(-w[x]\) -
              cos(\[Pi]\ x) + \((\[Pi]\^4 - 1)\)\ \(sin(\[Pi]\ x)\) + u(x) +
              v(x)\)}]}], ";"}], TraditionalForm]], "Input"],

Cell[TextData[{
  "Eliminate ",
  Cell[BoxData[
      \(TraditionalForm\`u(x)\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`\(u\^\[Prime]\)(x)\)]],
  ":"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{\(\[ScriptCapitalE]\_3\), "=",
        RowBox[{"Eliminate", "[",
          RowBox[{
            RowBox[{"{",

              RowBox[{\(\[ScriptCapitalE]\_1\), ",", \(\[ScriptCapitalE]\_2\),
                 ",",

                FractionBox[\(\[PartialD]\[ScriptCapitalE]\_2\), \
\(\[PartialD]x\),
                  MultilineFunction->None]}], "}"}], ",",
            RowBox[{"{",
              RowBox[{\(u(x)\), ",",
                RowBox[{
                  SuperscriptBox["u", "\[Prime]",
                    MultilineFunction->None], "(", "x", ")"}]}], "}"}]}],
          "]"}]}], TraditionalForm]], "Input"],

Cell[BoxData[
    FormBox[
      RowBox[{
        RowBox[{\(\[Pi]\^5\ \(cos(\[Pi]\ x)\)\),
          "-", \(\[Pi]\ \(cos(\[Pi]\ x)\)\),
          "+", \(\[Pi]\^4\ \(sin(\[Pi]\ x)\)\), "-", \(2\ \(sin(\[Pi]\ x)\)\),
           "+", \(2\ \(v(x)\)\), "+",
          RowBox[{
            SuperscriptBox["v", "\[Prime]",
              MultilineFunction->None], "(", "x", ")"}], "-",
          RowBox[{
            SuperscriptBox["v",
              TagBox[\((5)\),
                Derivative],
              MultilineFunction->None], "(", "x", ")"}]}], "==",
        RowBox[{
          SuperscriptBox["v",
            TagBox[\((4)\),
              Derivative],
            MultilineFunction->None], "(", "x", ")"}]}],
      TraditionalForm]], "Output"]
}, Open  ]],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " cannot solve this ",
  Cell[BoxData[
      \(TraditionalForm\`5\^th\)]],
  " order equation, and even the homogenous equation is non-trivial:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{"DSolve", "[",
        RowBox[{
          FormBox[
            RowBox[{
              RowBox[{\(2\ \(v(x)\)\), "+",
                RowBox[{
                  SuperscriptBox["v", "\[Prime]",
                    MultilineFunction->None], "(", "x", ")"}], "-",
                RowBox[{
                  SuperscriptBox["v",
                    TagBox[\((5)\),
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              RowBox[{
                SuperscriptBox["v",
                  TagBox[\((4)\),
                    Derivative],
                  MultilineFunction->None], "(", "x", ")"}]}],
            "TraditionalForm"], ",", \(v[x]\), ",", "x"}], "]"}],
      TraditionalForm]], "Input"],

Cell[BoxData[
    FormBox[
      RowBox[{"{",
        RowBox[{"{",
          RowBox[{\(v(x)\), "->",
            RowBox[{

              RowBox[{\(\[ExponentialE]\^\(x\ Root[#1\^5 + #1\^4 - #1 - 2 &,
                        1]\)\), " ",
                SubscriptBox[
                  TagBox["c",
                    C], "1"]}], "+",

              RowBox[{\(\[ExponentialE]\^\(x\ Root[#1\^5 + #1\^4 - #1 - 2 &,
                        2]\)\), " ",
                SubscriptBox[
                  TagBox["c",
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              RowBox[{\(\[ExponentialE]\^\(x\ Root[#1\^5 + #1\^4 - #1 - 2 &,
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                SubscriptBox[
                  TagBox["c",
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              RowBox[{\(\[ExponentialE]\^\(x\ Root[#1\^5 + #1\^4 - #1 - 2 &,
                        4]\)\), " ",
                SubscriptBox[
                  TagBox["c",
                    C], "4"]}], "+",

              RowBox[{\(\[ExponentialE]\^\(x\ Root[#1\^5 + #1\^4 - #1 - 2 &,
                        5]\)\), " ",
                SubscriptBox[
                  TagBox["c",
                    C], "5"]}]}]}], "}"}], "}"}],
      TraditionalForm]], "Output"]
}, Open  ]],

Cell[TextData[{
  "The reason that the roots of a ",
  Cell[BoxData[
      \(TraditionalForm\`5\^th\)]],
  " order polynomial arise can be seen from the auxilliary equation (found by \
putting ",
  Cell[BoxData[
      \(TraditionalForm\`v -> Function[x, \[ExponentialE]\^\(r\ x\)]\)]],
  "): "
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{
        RowBox[{\(\[ExponentialE]\^\(\(-r\)\ x\)\), " ",
          RowBox[{"(",
            RowBox[{
              RowBox[{
                RowBox[{\(2\ \(v(x)\)\), "+",
                  RowBox[{
                    SuperscriptBox["v", "\[Prime]",
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                    SuperscriptBox["v",
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                RowBox[{
                  SuperscriptBox["v",
                    TagBox[\((4)\),
                      Derivative],
                    MultilineFunction->None], "(", "x", ")"}]}],
              "/.", \(v -> Function[x, \[ExponentialE]\^\(r\ x\)]\)}],
            ")"}]}], "//", "Simplify"}], TraditionalForm]], "Input"],

Cell[BoxData[
    \(TraditionalForm\`r + 2 == r\^4\ \((r + 1)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm\`Solve[%, r]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`{{r -> Root[#1\^5 + #1\^4 - #1 - 2 &, 1]}, {r ->
          Root[#1\^5 + #1\^4 - #1 - 2 &, 2]}, {r ->
          Root[#1\^5 + #1\^4 - #1 - 2 &, 3]}, {r ->
          Root[#1\^5 + #1\^4 - #1 - 2 &, 4]}, {r ->
          Root[#1\^5 + #1\^4 - #1 - 2 &, 5]}}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "However, the form of the ",
  Cell[BoxData[
      \(TraditionalForm\`5\^th\)]],
  " order equation immediately suggests that the solution is of the form ",
  Cell[BoxData[
      \(TraditionalForm\`\[Alpha]\ \(sin(\[Pi]\ x)\) + \[Beta]\ \(cos(\[Pi]\ \
x)\)\)]],
  ":"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm\`\[ScriptCapitalE]\_3 /.
        v -> Function[
            x, \[Alpha]\ \(sin(\[Pi]\ x)\) + \[Beta]\ \(cos(\[Pi]\ x)\)] //
      Simplify\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`\((\(-\[Pi]\^5\)\ \((\[Alpha] -
                    1)\) + \[Pi]\ \((\[Alpha] - 1)\) - \[Pi]\^4\ \[Beta] +
              2\ \[Beta])\)\ \(cos(\[Pi]\ x)\) + \((\(-\[Pi]\^4\)\ \[Alpha] +
              2\ \[Alpha] + \[Pi]\^5\ \[Beta] - \[Pi]\ \[Beta] + \[Pi]\^4 -
              2)\)\ \(sin(\[Pi]\ x)\) == 0\)], "Output"]
}, Open  ]],

Cell["It is easy to compute the undetermined coefficients,", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm\`Solve[{Coefficient[%, sin(\[Pi]\ x)],
        Coefficient[%, cos(\[Pi]\ x)]}, {\[Alpha], \[Beta]}]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`{{\[Alpha] -> 1, \[Beta] -> 0}}\)], "Output"]
}, Open  ]],

Cell["\<\
and verify that the solution satisfies the boundary \
conditions.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{
        RowBox[{"{",
          RowBox[{\(v(0) == 0\), ",",
            RowBox[{
              RowBox[{
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            RowBox[{
              RowBox[{
                SuperscriptBox["v", "\[Prime]\[Prime]",
                  MultilineFunction->None], "(", "1", ")"}], "==", "0"}]}],
          "}"}], "/.", \(v -> Function[x, sin(\[Pi]\ x)]\)}],
      TraditionalForm]], "Input"],

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

Cell["Now solving the remaining first-order equation is trivial.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm\`DSolve[{\[ScriptCapitalE]\_1 /.
            v -> Function[x, sin(\[Pi]\ x)], u[0] == 2}, u[x], x] //
      Simplify\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`{{u(x) ->
          cos(\[Pi]\ x) + \[ExponentialE]\^\(\(-a\)\ x\)}}\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Mathematica solution", "Section"],

Cell[TextData[{
  "The ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " solution to the original pair of equations is rather unenlightening:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm\`DSolve[{\[ScriptCapitalE]\_1, \[ScriptCapitalE]\_2}, \
{u(x), v(x)}, x]\)], "Input"],

Cell[BoxData[
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      RowBox[{"{",
        RowBox[{"{",
          RowBox[{
            RowBox[{\(u(x)\), "->",
              RowBox[{
                RowBox[{
                  RowBox[{"(",
                    RowBox[{
                      SubsuperscriptBox["\[Integral]",
                        SubscriptBox[
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                        "x"], \(\((RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\^3 - \[ExponentialE]\^\(\(-DSolve`t\)\ #1\)\ #1\^2 + \[ExponentialE]\
\^\(\(-DSolve`t\)\ #1\)\ #1 - \[ExponentialE]\^\(\(-DSolve`t\)\ #1\)\)\/\(5\ \
#1\^3 - #1\^2 + #1 - 1\) &]\ \((\(-\[ExponentialE]\^\(\(-a\)\ DSolve`t\)\)\ a \
+ \[ExponentialE]\^\(\(-a\)\ DSolve`t\) +
                                  cos(\[Pi]\ DSolve`t) - \[Pi]\ \(sin(\[Pi]\ \
DSolve`t)\) - sin(\[Pi]\ DSolve`t))\) +
                            RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \[ExponentialE]\^\(\(-DSolve`t\)\ #1\
\)\/\(5\ #1\^4 + 4\ #1\^3 - 1\) &]\ \((\(-\(cos(\[Pi]\ DSolve`t)\)\) - \
\[ExponentialE]\^\(\(-a\)\ DSolve`t\) + \[Pi]\^4\ \(sin(\[Pi]\ DSolve`t)\) -
                                  sin(\[Pi]\ DSolve`t))\))\) \
\[DifferentialD]DSolve`t\)}], ")"}],
                  " ", \(RootSum[#1\^5 + #1\^4 - #1 -
                        2 &, \(\[ExponentialE]\^\(x\ #1\)\ #1\^3 - \
\[ExponentialE]\^\(x\ #1\)\ #1\^2 + \[ExponentialE]\^\(x\ #1\)\ #1 - \
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                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\^3\)\/\(5\ #1\^4 + 4\ #1\^3 - 1\) &]\ \
\((\(-\[ExponentialE]\^\(\(-a\)\ DSolve`t\)\)\ a + \[ExponentialE]\^\(\(-a\)\ \
DSolve`t\) + cos(\[Pi]\ DSolve`t) - \[Pi]\ \(sin(\[Pi]\ DSolve`t)\) -
                                  sin(\[Pi]\ DSolve`t))\) +
                            RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\^3\)\/\(5\ #1\^3 - #1\^2 + #1 - 1\) &]\ \((\(-\(cos(\[Pi]\ DSolve`t)\
\)\) - \[ExponentialE]\^\(\(-a\)\ DSolve`t\) + \[Pi]\^4\ \(sin(\[Pi]\ \
DSolve`t)\) - sin(\[Pi]\ DSolve`t))\))\) \[DifferentialD]DSolve`t\)}], ")"}],
                  " ", \(RootSum[#1\^5 + #1\^4 - #1 -
                        2 &, \[ExponentialE]\^\(x\ #1\)\/\(5\ #1\^4 + 4\ \
#1\^3 - 1\) &]\)}], "+",
                RowBox[{
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                      SubsuperscriptBox["\[Integral]",
                        SubscriptBox[
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                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\^2\)\/\(5\ #1\^4 + 4\ #1\^3 - 1\) &]\ \
\((\(-\[ExponentialE]\^\(\(-a\)\ DSolve`t\)\)\ a + \[ExponentialE]\^\(\(-a\)\ \
DSolve`t\) + cos(\[Pi]\ DSolve`t) - \[Pi]\ \(sin(\[Pi]\ DSolve`t)\) -
                                  sin(\[Pi]\ DSolve`t))\) +
                            RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\^2\)\/\(5\ #1\^3 - #1\^2 + #1 - 1\) &]\ \((\(-\(cos(\[Pi]\ DSolve`t)\
\)\) - \[ExponentialE]\^\(\(-a\)\ DSolve`t\) + \[Pi]\^4\ \(sin(\[Pi]\ \
DSolve`t)\) - sin(\[Pi]\ DSolve`t))\))\) \[DifferentialD]DSolve`t\)}], ")"}],
                  " ", \(RootSum[#1\^5 + #1\^4 - #1 -
                        2 &, \(\[ExponentialE]\^\(x\ #1\)\ #1\)\/\(5\ #1\^4 + \
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                      SubsuperscriptBox["\[Integral]",
                        SubscriptBox[
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                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\)\/\(5\ #1\^4 + 4\ #1\^3 - 1\) &]\ \((\(-\[ExponentialE]\^\(\(-a\)\ \
DSolve`t\)\)\ a + \[ExponentialE]\^\(\(-a\)\ DSolve`t\) +
                                  cos(\[Pi]\ DSolve`t) - \[Pi]\ \(sin(\[Pi]\ \
DSolve`t)\) - sin(\[Pi]\ DSolve`t))\) +
                            RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\)\/\(5\ #1\^3 - #1\^2 + #1 - 1\) &]\ \((\(-\(cos(\[Pi]\ \
DSolve`t)\)\) - \[ExponentialE]\^\(\(-a\)\ DSolve`t\) + \[Pi]\^4\ \(sin(\[Pi]\
\ DSolve`t)\) - sin(\[Pi]\ DSolve`t))\))\) \[DifferentialD]DSolve`t\)}],
                    ")"}], " ", \(RootSum[#1\^5 + #1\^4 - #1 -
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                      SubsuperscriptBox["\[Integral]",
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\)\/\(5\ #1\^4 + 4\ #1\^3 - 1\) &]\ \((\(-\[ExponentialE]\^\(\(-a\)\ DSolve`t\
\)\)\ a + \[ExponentialE]\^\(\(-a\)\ DSolve`t\) +
                                  cos(\[Pi]\ DSolve`t) - \[Pi]\ \(sin(\[Pi]\ \
DSolve`t)\) - sin(\[Pi]\ DSolve`t))\) +
                            RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \[ExponentialE]\^\(\(-DSolve`t\)\ #1\
\)\/\(5\ #1\^3 - #1\^2 + #1 - 1\) &]\ \((\(-\(cos(\[Pi]\ DSolve`t)\)\) - \
\[ExponentialE]\^\(\(-a\)\ DSolve`t\) + \[Pi]\^4\ \(sin(\[Pi]\ DSolve`t)\) -
                                  sin(\[Pi]\ DSolve`t))\))\) \
\[DifferentialD]DSolve`t\)}], ")"}],
                  " ", \(RootSum[#1\^5 + #1\^4 - #1 -
                        2 &, \(\[ExponentialE]\^\(x\ #1\)\ #1\^3\)\/\(5\ \
#1\^4 + 4\ #1\^3 - 1\) &]\)}]}]}], ",",
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                RowBox[{
                  RowBox[{"(",
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                      SubsuperscriptBox["\[Integral]",
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                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\^3\)\/\(5\ #1\^4 + 4\ #1\^3 - 1\) &]\ \
\((\(-\[ExponentialE]\^\(\(-a\)\ DSolve`t\)\)\ a + \[ExponentialE]\^\(\(-a\)\ \
DSolve`t\) + cos(\[Pi]\ DSolve`t) - \[Pi]\ \(sin(\[Pi]\ DSolve`t)\) -
                                  sin(\[Pi]\ DSolve`t))\) +
                            RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\^3\)\/\(5\ #1\^3 - #1\^2 + #1 - 1\) &]\ \((\(-\(cos(\[Pi]\ DSolve`t)\
\)\) - \[ExponentialE]\^\(\(-a\)\ DSolve`t\) + \[Pi]\^4\ \(sin(\[Pi]\ \
DSolve`t)\) - sin(\[Pi]\ DSolve`t))\))\) \[DifferentialD]DSolve`t\)}], ")"}],
                  " ", \(RootSum[#1\^5 + #1\^4 - #1 -
                        2 &, \[ExponentialE]\^\(x\ #1\)\/\(5\ #1\^3 - #1\^2 + \
#1 - 1\) &]\)}], "+",
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                      SubsuperscriptBox["\[Integral]",
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                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\^2\)\/\(5\ #1\^4 + 4\ #1\^3 - 1\) &]\ \
\((\(-\[ExponentialE]\^\(\(-a\)\ DSolve`t\)\)\ a + \[ExponentialE]\^\(\(-a\)\ \
DSolve`t\) + cos(\[Pi]\ DSolve`t) - \[Pi]\ \(sin(\[Pi]\ DSolve`t)\) -
                                  sin(\[Pi]\ DSolve`t))\) +
                            RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\^2\)\/\(5\ #1\^3 - #1\^2 + #1 - 1\) &]\ \((\(-\(cos(\[Pi]\ DSolve`t)\
\)\) - \[ExponentialE]\^\(\(-a\)\ DSolve`t\) + \[Pi]\^4\ \(sin(\[Pi]\ \
DSolve`t)\) - sin(\[Pi]\ DSolve`t))\))\) \[DifferentialD]DSolve`t\)}], ")"}],
                  " ", \(RootSum[#1\^5 + #1\^4 - #1 -
                        2 &, \(\[ExponentialE]\^\(x\ #1\)\ #1\)\/\(5\ #1\^3 - \
#1\^2 + #1 - 1\) &]\)}], "+",
                RowBox[{
                  RowBox[{"(",
                    RowBox[{
                      SubsuperscriptBox["\[Integral]",
                        SubscriptBox[
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                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\)\/\(5\ #1\^4 + 4\ #1\^3 - 1\) &]\ \((\(-\[ExponentialE]\^\(\(-a\)\ \
DSolve`t\)\)\ a + \[ExponentialE]\^\(\(-a\)\ DSolve`t\) +
                                  cos(\[Pi]\ DSolve`t) - \[Pi]\ \(sin(\[Pi]\ \
DSolve`t)\) - sin(\[Pi]\ DSolve`t))\) +
                            RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\)\/\(5\ #1\^3 - #1\^2 + #1 - 1\) &]\ \((\(-\(cos(\[Pi]\ \
DSolve`t)\)\) - \[ExponentialE]\^\(\(-a\)\ DSolve`t\) + \[Pi]\^4\ \(sin(\[Pi]\
\ DSolve`t)\) - sin(\[Pi]\ DSolve`t))\))\) \[DifferentialD]DSolve`t\)}],
                    ")"}], " ", \(RootSum[#1\^5 + #1\^4 - #1 -
                        2 &, \(\[ExponentialE]\^\(x\ #1\)\ #1\^2\)\/\(5\ \
#1\^3 - #1\^2 + #1 - 1\) &]\)}], "+",
                RowBox[{
                  RowBox[{"(",
                    RowBox[{
                      SubsuperscriptBox["\[Integral]",
                        SubscriptBox[
                          TagBox["c",
                            C], "2"],
                        "x"], \(\((RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \[ExponentialE]\^\(\(-DSolve`t\)\ #1\
\)\/\(5\ #1\^4 + 4\ #1\^3 - 1\) &]\ \((\(-\[ExponentialE]\^\(\(-a\)\ DSolve`t\
\)\)\ a + \[ExponentialE]\^\(\(-a\)\ DSolve`t\) +
                                  cos(\[Pi]\ DSolve`t) - \[Pi]\ \(sin(\[Pi]\ \
DSolve`t)\) - sin(\[Pi]\ DSolve`t))\) +
                            RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \[ExponentialE]\^\(\(-DSolve`t\)\ #1\
\)\/\(5\ #1\^3 - #1\^2 + #1 - 1\) &]\ \((\(-\(cos(\[Pi]\ DSolve`t)\)\) - \
\[ExponentialE]\^\(\(-a\)\ DSolve`t\) + \[Pi]\^4\ \(sin(\[Pi]\ DSolve`t)\) -
                                  sin(\[Pi]\ DSolve`t))\))\) \
\[DifferentialD]DSolve`t\)}], ")"}],
                  " ", \(RootSum[#1\^5 + #1\^4 - #1 -
                        2 &, \(\[ExponentialE]\^\(x\ #1\)\ #1\^3\)\/\(5\ \
#1\^3 - #1\^2 + #1 - 1\) &]\)}], "+",
                RowBox[{
                  RowBox[{"(",
                    RowBox[{
                      SubsuperscriptBox["\[Integral]",
                        SubscriptBox[
                          TagBox["c",
                            C], "1"],
                        "x"], \(\((RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \(\[ExponentialE]\^\(\(-DSolve`t\)\ \
#1\)\ #1\^3 - \[ExponentialE]\^\(\(-DSolve`t\)\ #1\)\ #1\^2 + \[ExponentialE]\
\^\(\(-DSolve`t\)\ #1\)\ #1 - \[ExponentialE]\^\(\(-DSolve`t\)\ #1\)\)\/\(5\ \
#1\^3 - #1\^2 + #1 - 1\) &]\ \((\(-\[ExponentialE]\^\(\(-a\)\ DSolve`t\)\)\ a \
+ \[ExponentialE]\^\(\(-a\)\ DSolve`t\) +
                                  cos(\[Pi]\ DSolve`t) - \[Pi]\ \(sin(\[Pi]\ \
DSolve`t)\) - sin(\[Pi]\ DSolve`t))\) +
                            RootSum[#1\^5 + #1\^4 - #1 -
                                    2 &, \[ExponentialE]\^\(\(-DSolve`t\)\ #1\
\)\/\(5\ #1\^4 + 4\ #1\^3 - 1\) &]\ \((\(-\(cos(\[Pi]\ DSolve`t)\)\) - \
\[ExponentialE]\^\(\(-a\)\ DSolve`t\) + \[Pi]\^4\ \(sin(\[Pi]\ DSolve`t)\) -
                                  sin(\[Pi]\ DSolve`t))\))\) \
\[DifferentialD]DSolve`t\)}], ")"}],
                  " ", \(RootSum[#1\^5 + #1\^4 - #1 -
                        2 &, \[ExponentialE]\^\(x\ #1\)\/\(5\ #1\^4 + 4\ \
#1\^3 - 1\) &]\)}]}]}]}], "}"}], "}"}], TraditionalForm]], "Output"]
}, Closed]]
}, Closed]]
}
]


____________________________________________________________________
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia
Nedlands WA  6907                     mailto:paul at physics.uwa.edu.au
AUSTRALIA                            http://physics.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________




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