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(* Beginning of Notebook Content *)
Notebook[{
Cell[TextData[StyleBox["Use the package IntroToSymmetry.m to work out the \
point group of the second order ODE\n\nYxx+Yx^2/Y-1/Y=0 \n\nconsidered in \
homework exercise 8.9.",
 FontSize->14,
 FontVariations->{"CompatibilityType"->0}]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Clear all symbols in the current context.", "Text",
 CellChangeTimes->{{3.4715644485587473`*^9, 3.47156448605101*^9}}],

Cell[BoxData[
 RowBox[{"ClearAll", "[", 
  RowBox[{"Evaluate", "[", 
   RowBox[{
    RowBox[{"Context", "[", "]"}], "<>", "\"\<*\>\""}], "]"}], "]"}]], "Input"],

Cell[TextData[{
 StyleBox["First read in the package which is located in User Home \
Folder/Library/",
  FontWeight->"Plain"],
 StyleBox["Mathematica",
  FontWeight->"Plain",
  FontSlant->"Italic"],
 StyleBox["/Applications/SymmetryAnalysis.",
  FontWeight->"Plain"]
}], "Text",
 CellChangeTimes->{{3.472046215957556*^9, 3.472046255573135*^9}, {
  3.472046289795615*^9, 3.472046308115048*^9}, {3.4720469527121572`*^9, 
  3.4720469532543707`*^9}},
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Needs[\"SymmetryAnalysis`IntroToSymmetry`\"]", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Enter the input equation as a string, ie, in quotes.", "Text"],

Cell["\<\
inputequation=
\"D[y[x],x,x]+D[y[x],x]^2/y[x]-1/y[x]\";\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["\<\
The function y[x] is a solution of the equation and this constraint must be \
applied to the invariance condition. Replace the second derivative wherever \
it appears in the invariance condition.\
\>", "Text"],

Cell["\<\
rulesarray=
{\"D[y[x],x,x]->-D[y[x],x]^2/y[x]+1/y[x]\"};\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[TextData[StyleBox["Enter the list of independent variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"independentvariables", "=", 
   RowBox[{"{", "\"\<x\>\"", "}"}]}], ";"}]], "Input"],

Cell[TextData[StyleBox["Enter the list of dependent variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"dependentvariables", "=", 
   RowBox[{"{", "\"\<y\>\"", "}"}]}], ";"}]], "Input"],

Cell[TextData[StyleBox["Enter the list of function names and constants that \
need to be protected when the equation is converted to generic (x1,y1) \
variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"frozennames", "=", 
   RowBox[{"{", "}"}]}], ";"}]], "Input"],

Cell[TextData[StyleBox["Enter the maximum derivative order of the equation.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"p", "=", "2"}], ";"}]], "Input"],

Cell[TextData[StyleBox["The maximum derivative order that the infinitesimals \
are assumed to depend on is specified by the input parameter r. This \
parameter is only nonzero when the user is looking for Lie contact groups or \
Lie-Backlund groups. For the usual case where one is searching for point \
groups one sets r=0.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"r", "=", "0"}], ";"}]], "Input"],

Cell[TextData[StyleBox["When searching for Lie-Backlund groups (r=1 or \
greater) one can, without loss of generality, leave the independent variables \
untransformed. The corresponding infinitesimals (the xse's) are set to zero \
by setting xseon=0. If one is searching for point groups then set xseon=1. \
The choice xseon=1 is also an option when looking for Lie-Backlund groups and \
this can be useful when looking for contact symmetries.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"xseon", "=", "1"}], ";"}]], "Input"],

Cell[TextData[StyleBox["When searching for Lie-Backlund groups, it is \
necessary to differentiate the input equation r times producing derivatives \
of order p+r. These higher order differential consequences are appended to \
the set of rules applied to the invariance condition. This process is carried \
out automatically when internalrules=1. For point groups the equation or \
equation system is the only rule or set of rules needed and one sets \
internalrules=0.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"internalrules", "=", "0"}], ";"}]], "Input"],

Cell[TextData[StyleBox["Now work out the determining equations of the Lie \
point group that leaves the equation invariant. ",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[CellGroupData[{

Cell["\<\
Timing[FindDeterminingEquations[
independentvariables,dependentvariables,frozennames,p,r,xseon,
inputequation,rulesarray,internalrules]]\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 InterpretationBox["\<\"The function FindDetermining Equations has begun, the \
memory in use = \\!\\(26716944\\), the time used = \\!\\(11.168928`\\)\"\>",
  StringForm[
  "The function FindDetermining Equations has begun, the memory in use = ``, \
the time used = ``", 26716944, 11.168928],
  Editable->False]], "Print",
 CellChangeTimes->{3.469399466041264*^9, 3.469399662377552*^9, 
  3.471554493112694*^9, 3.471576919601972*^9, 3.483555833044691*^9}],

Cell[BoxData["\<\"The function FindDeterminingEquations is nearly complete. \
The invariance condition has been created with all rules applied. The final \
step in the generation of the determining equations is to sum together terms \
in the table of invariance condition terms (called infinitesimaltable) that \
are multiplied by the same combination of products of free y derivatives. The \
result is the table infinitesimaltablesums corresponding to matching \
y-derivative expressions. If the invariance condition is long as it often is \
this process could take a long time since it requires sorting through the \
table infinitesimaltable once for each possible combination of y derivative \
products. This is the rate limiting step in the function \
FindDeterminingEquations.Virtually all other steps are quite fast including \
the generation of the extended derivatives of the infinitesimals.\"\>"], \
"Print",
 CellChangeTimes->{3.469399466041264*^9, 3.469399662377552*^9, 
  3.471554493112694*^9, 3.471576919601972*^9, 3.483555833102749*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The determining equations have been expressed in \
terms of z-variables, the length of zdeterminingequations = \\!\\(4\\), the \
byte count of zdeterminingequations = \\!\\(704\\), the memory in use = \
\\!\\(26747312\\), the time used = \\!\\(11.210802`\\)\"\>",
  StringForm[
  "The determining equations have been expressed in terms of z-variables, the \
length of zdeterminingequations = ``, the byte count of zdeterminingequations \
= ``, the memory in use = ``, the time used = ``", 4, 704, 26747312, 
   11.210802],
  Editable->False]], "Print",
 CellChangeTimes->{3.469399466041264*^9, 3.469399662377552*^9, 
  3.471554493112694*^9, 3.471576919601972*^9, 3.4835558331052437`*^9}],

Cell[BoxData[
 InterpretationBox["\<\"FindDeterminingEquations is done. The memory in use = \
\\!\\(26748136\\), the time used = \\!\\(11.210993`\\)\"\>",
  StringForm[
  "FindDeterminingEquations is done. The memory in use = ``, the time used = \
``", 26748136, 11.210993],
  Editable->False]], "Print",
 CellChangeTimes->{3.469399466041264*^9, 3.469399662377552*^9, 
  3.471554493112694*^9, 3.471576919601972*^9, 3.483555833107102*^9}],

Cell[BoxData["\<\"FindDeterminingEquations has finished executing. You can \
look at the output in the table zdeterminingequations. Each entry in this \
table is a determining equation in string format expressed in terms of \
z-variables. Rules for converting between z-variables and conventional \
variables are contained in the table ztableofrules. To view the determining \
equations in terms of conventional variables use the command \
ToExpression[zdeterminingequations]/.ztableofrules. There are two other items \
the user may wish to look at; the equation converted to generic \
(x1,x2,...,y1,y2,...) variables is designated equationgenericvariables and \
the various derivatives of the equation that appear in the invariance \
condition can be viewed in the table invarconditiontable. Rules for \
converting between z-variables and generic variables are contained in the \
table ztableofrulesxy.\"\>"], "Print",
 CellChangeTimes->{3.469399466041264*^9, 3.469399662377552*^9, 
  3.471554493112694*^9, 3.471576919601972*^9, 3.483555833108821*^9}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0.04276899999999806`", ",", "Null"}], "}"}]], "Output",
 CellChangeTimes->{3.469399466079534*^9, 3.4693996624143143`*^9, 
  3.471554493214549*^9, 3.471576919637773*^9, 3.483555833111231*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData["equationgenericvariables"], "Input"],

Cell[BoxData["\<\"D[y1[x1],x1,x1]+D[y1[x1],x1]^2/y1[x1]-1/y1[x1]\"\>"], \
"Output",
 CellChangeTimes->{3.469399466111815*^9, 3.469399662476348*^9, 
  3.471554493265115*^9, 3.4715769196709213`*^9, 3.483555833145934*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData["invarconditiontable"], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0", ",", 
   RowBox[{
    FractionBox["1", 
     SuperscriptBox[
      RowBox[{"y1", "[", "x1", "]"}], "2"]], "-", 
    FractionBox[
     SuperscriptBox[
      RowBox[{
       SuperscriptBox["y1", "\[Prime]",
        MultilineFunction->None], "[", "x1", "]"}], "2"], 
     SuperscriptBox[
      RowBox[{"y1", "[", "x1", "]"}], "2"]]}], ",", 
   FractionBox[
    RowBox[{"2", " ", 
     RowBox[{
      SuperscriptBox["y1", "\[Prime]",
       MultilineFunction->None], "[", "x1", "]"}]}], 
    RowBox[{"y1", "[", "x1", "]"}]], ",", "1"}], "}"}]], "Output",
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  3.471554493323237*^9, 3.471576919725873*^9, 3.483555833184417*^9}]
}, Open  ]],

Cell["Look at the determining equations", "Text"],

Cell[CellGroupData[{

Cell["zdeterminingequations", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"\<\"z2*Derivative[0, 1][xse1][z1, z2] - z2^2*Derivative[0, \
2][xse1][z1, z2] == 0\"\>", 
   ",", "\<\"-eta1[z1, z2] + z2*Derivative[0, 1][eta1][z1, z2] + \
z2^2*Derivative[0, 2][eta1][z1, z2] - 2*z2^2*Derivative[1, 1][xse1][z1, z2] \
== 0\"\>", 
   ",", "\<\"eta1[z1, z2] + z2*Derivative[0, 1][eta1][z1, z2] - \
2*z2*Derivative[1, 0][xse1][z1, z2] + z2^2*Derivative[2, 0][eta1][z1, z2] == \
0\"\>", ",", "\<\"-3*z2*Derivative[0, 1][xse1][z1, z2] + 2*z2*Derivative[1, \
0][eta1][z1, z2] + 2*z2^2*Derivative[1, 1][eta1][z1, z2] - z2^2*Derivative[2, \
0][xse1][z1, z2] == 0\"\>"}], "}"}]], "Output",
 CellChangeTimes->{3.4693994661939707`*^9, 3.469399662547392*^9, 
  3.4715544933995943`*^9, 3.4715769197719517`*^9, 3.483555833217978*^9}]
}, Open  ]],

Cell["\<\
Here are the determining equations in a slightly more readable form as \
expressions rather than strings.\
\>", "Text"],

Cell[CellGroupData[{

Cell["zdeteqns=ToExpression[zdeterminingequations]", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"z2", " ", 
      RowBox[{
       SuperscriptBox["xse1", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"0", ",", "1"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}], "-", 
     RowBox[{
      SuperscriptBox["z2", "2"], " ", 
      RowBox[{
       SuperscriptBox["xse1", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"0", ",", "2"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}]}], "\[Equal]", "0"}], ",", 
   RowBox[{
    RowBox[{
     RowBox[{"-", 
      RowBox[{"eta1", "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}], "+", 
     RowBox[{"z2", " ", 
      RowBox[{
       SuperscriptBox["eta1", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"0", ",", "1"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}], "+", 
     RowBox[{
      SuperscriptBox["z2", "2"], " ", 
      RowBox[{
       SuperscriptBox["eta1", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"0", ",", "2"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}], "-", 
     RowBox[{"2", " ", 
      SuperscriptBox["z2", "2"], " ", 
      RowBox[{
       SuperscriptBox["xse1", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"1", ",", "1"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}]}], "\[Equal]", "0"}], ",", 
   RowBox[{
    RowBox[{
     RowBox[{"eta1", "[", 
      RowBox[{"z1", ",", "z2"}], "]"}], "+", 
     RowBox[{"z2", " ", 
      RowBox[{
       SuperscriptBox["eta1", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"0", ",", "1"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}], "-", 
     RowBox[{"2", " ", "z2", " ", 
      RowBox[{
       SuperscriptBox["xse1", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"1", ",", "0"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}], "+", 
     RowBox[{
      SuperscriptBox["z2", "2"], " ", 
      RowBox[{
       SuperscriptBox["eta1", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"2", ",", "0"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}]}], "\[Equal]", "0"}], ",", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"-", "3"}], " ", "z2", " ", 
      RowBox[{
       SuperscriptBox["xse1", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"0", ",", "1"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}], "+", 
     RowBox[{"2", " ", "z2", " ", 
      RowBox[{
       SuperscriptBox["eta1", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"1", ",", "0"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}], "+", 
     RowBox[{"2", " ", 
      SuperscriptBox["z2", "2"], " ", 
      RowBox[{
       SuperscriptBox["eta1", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"1", ",", "1"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}], "-", 
     RowBox[{
      SuperscriptBox["z2", "2"], " ", 
      RowBox[{
       SuperscriptBox["xse1", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"2", ",", "0"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"z1", ",", "z2"}], "]"}]}]}], "\[Equal]", "0"}]}], 
  "}"}]], "Output",
 CellChangeTimes->{3.4693994662499113`*^9, 3.469399662604033*^9, 
  3.471554493455257*^9, 3.4715769198277683`*^9, 3.4835558332518663`*^9}]
}, Open  ]],

Cell["\<\
Here is the correspondence between z-variables and conventional variables.\
\>", "Text"],

Cell[CellGroupData[{

Cell["ztableofrules", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"z1", "\[Rule]", "x"}], ",", 
   RowBox[{"z2", "\[Rule]", 
    RowBox[{"y", "[", "x", "]"}]}]}], "}"}]], "Output",
 CellChangeTimes->{3.469399466282628*^9, 3.469399662635839*^9, 
  3.4715544934991817`*^9, 3.471576919871512*^9, 3.483555833284443*^9}]
}, Open  ]],

Cell["\<\
Now solve the determining equations in terms of multivariable polynomials of \
a selected order.\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
Timing[SolveDeterminingEquations[
independentvariables,dependentvariables,r,
xseon,zdeterminingequations,order=3]]\
\>", "Input",
 CellChangeTimes->{{3.4693994412853613`*^9, 3.469399441739595*^9}},
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 InterpretationBox["\<\"The variable powertablelength is the number of terms \
required for each multivariate polynomial used for the infinitesimals. This \
number is determined by the choice of polynomial order and the number of \
zvariables. The time needed to solve the determining equations increases as \
powertable increases. powertablelength = \\!\\(10\\)\"\>",
  StringForm[
  "The variable powertablelength is the number of terms required for each \
multivariate polynomial used for the infinitesimals. This number is \
determined by the choice of polynomial order and the number of zvariables. \
The time needed to solve the determining equations increases as powertable \
increases. powertablelength = ``", 10],
  Editable->False]], "Print",
 CellChangeTimes->{3.469399466328171*^9, 3.469399662681782*^9, 
  3.4715544935541983`*^9, 3.471576919927662*^9, 3.4835558333182297`*^9}],

Cell[BoxData["\<\"The polynomial expansions have been substituted into the \
determining equations. It is now time to collect the coefficients of various \
powers of zvariables into a table called table of coefficientsall. This step \
uses the function CoefficientList and is a fairly time consuming procedure.\"\
\>"], "Print",
 CellChangeTimes->{3.469399466328171*^9, 3.469399662681782*^9, 
  3.4715544935541983`*^9, 3.471576919927662*^9, 3.483555833320169*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(26777032\\), The time = \\!\
\\(11.224833000000002`\\)\"\>",
  StringForm[
  "The memory in use = ``, The time = ``", 26777032, 11.224833000000002`],
  Editable->False]], "Print",
 CellChangeTimes->{3.469399466328171*^9, 3.469399662681782*^9, 
  3.4715544935541983`*^9, 3.471576919927662*^9, 3.483555833321539*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The number of unknown polynomial coefficients = \
\\!\\(20\\)\"\>",
  StringForm["The number of unknown polynomial coefficients = ``", 20],
  Editable->False]], "Print",
 CellChangeTimes->{3.469399466328171*^9, 3.469399662681782*^9, 
  3.4715544935541983`*^9, 3.471576919927662*^9, 3.483555833322688*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The number of equations for the polynomial \
coefficients = \\!\\(27\\)\"\>",
  StringForm[
  "The number of equations for the polynomial coefficients = ``", 27],
  Editable->False]], "Print",
 CellChangeTimes->{3.469399466328171*^9, 3.469399662681782*^9, 
  3.4715544935541983`*^9, 3.471576919927662*^9, 3.4835558333238287`*^9}],

Cell[BoxData["\<\"Now it we are ready to use the function Solve to find the \
nonzero polynomial coefficients corresponding to the symmetries of the input \
equation(s). This can take a while.\"\>"], "Print",
 CellChangeTimes->{3.469399466328171*^9, 3.469399662681782*^9, 
  3.4715544935541983`*^9, 3.471576919927662*^9, 3.4835558333249273`*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(26783976\\), The time = \\!\
\\(11.225977`\\)\"\>",
  StringForm["The memory in use = ``, The time = ``", 26783976, 11.225977],
  Editable->False]], "Print",
 CellChangeTimes->{3.469399466328171*^9, 3.469399662681782*^9, 
  3.4715544935541983`*^9, 3.471576919927662*^9, 3.483555833326152*^9}],

Cell[BoxData["\<\"Solve has finished.\"\>"], "Print",
 CellChangeTimes->{3.469399466328171*^9, 3.469399662681782*^9, 
  3.4715544935541983`*^9, 3.471576919927662*^9, 3.48355583332724*^9}],

Cell[BoxData["\<\"The function SolveDeterminingEquations is finished \
executing.\"\>"], "Print",
 CellChangeTimes->{3.469399466328171*^9, 3.469399662681782*^9, 
  3.4715544935541983`*^9, 3.471576919927662*^9, 3.483555833328285*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(26791336\\), The time = \\!\
\\(11.227027000000001`\\)\"\>",
  StringForm[
  "The memory in use = ``, The time = ``", 26791336, 11.227027000000001`],
  Editable->False]], "Print",
 CellChangeTimes->{3.469399466328171*^9, 3.469399662681782*^9, 
  3.4715544935541983`*^9, 3.471576919927662*^9, 3.483555833329365*^9}],

Cell[BoxData["\<\"You can look at the output in the tables xsefunctions and \
etafunctions. Each entry in these tables is an infinitesimal function in \
string format expressed in terms of z-variables and the group parameters. The \
output can also be viewed with the group parameters stripped away by looking \
at the table infinitesimalgroups. In either case you may wish to convert the \
z-variables to conventional variables using the table ztableofrules. \
\\n\\nKeep in mind that this function only finds solutions of the determining \
equations that are of polynomial form. The determining equations may admit \
solutions that involve transcendental functions and/or integrals. Note that \
arbitrary functions may appear in the infinitesimals and that these can be \
detected by running the package function SolveDeterminingEquations for \
several polynomial orders. If terms of ever increasing order appear, then an \
arbitrary function is indicated.\"\>"], "Print",
 CellChangeTimes->{3.469399466328171*^9, 3.469399662681782*^9, 
  3.4715544935541983`*^9, 3.471576919927662*^9, 3.483555833330442*^9}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0.010856999999999672`", ",", "Null"}], "}"}]], "Output",
 CellChangeTimes->{3.469399466331674*^9, 3.469399662685203*^9, 
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