(* Content-type: application/mathematica *)

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(* http://www.wolfram.com/nb *)

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(* Beginning of Notebook Content *)
Notebook[{
Cell[TextData[StyleBox["In this example we use the package IntroToSymmetry.m \
to work out the groups for the second-order ODE \n\nYxx-Yx-Y^3=0 \n",
 FontSize->14,
 FontWeight->"Plain"]], "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}, {3.4720466725423717`*^9, 
  3.472046673357225*^9}},
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{"Needs", "[", "\"\<SymmetryAnalysis`IntroToSymmetry`\>\"", 
  "]"}]], "Input"],

Cell[TextData[StyleBox["Enter the input equation as a string. The \
correspondence between variables is x->x1, Y->y1. Don't include the ==0 at \
the end.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{
  "inputequation", "=", "\n", "\"\<D[y[x],x,x]-D[y[x],x]-y[x]^3\>\""}], 
  ";"}]], "Input"],

Cell[TextData[StyleBox["The function y[x] is a solution of the equation and \
this constraint must be applied in the form of a rule to the invariance \
condition. ",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"rulesarray", "=", "\[IndentingNewLine]", 
   RowBox[{"{", "\"\<D[y[x],x,x]->D[y[x],x]+y[x]^3\>\"", "}"}]}], 
  ";"}]], "Input"],

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 and constant names that \
must be preserved when the equations are expressed in terms of (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 set 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 \
orequation 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. The output is available as a \
table of strings called zdeterminingequations.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Timing", "[", 
  RowBox[{"FindDeterminingEquations", "[", "\[IndentingNewLine]", 
   RowBox[{
   "independentvariables", ",", "dependentvariables", ",", "frozennames", ",",
     "p", ",", "r", ",", "xseon", ",", "inputequation", ",", "rulesarray", 
    ",", "internalrules"}], "]"}], "]"}]], "Input"],

Cell[BoxData[
 InterpretationBox["\<\"The function FindDetermining Equations has begun, the \
memory in use = \\!\\(30534880\\), the time used = \
\\!\\(9.676594999999999`\\)\"\>",
  StringForm[
  "The function FindDetermining Equations has begun, the memory in use = ``, \
the time used = ``", 30534880, 9.676594999999999],
  Editable->False]], "Print",
 CellChangeTimes->{3.4694020398368797`*^9, 3.471554778448366*^9, 
  3.471577255865014*^9, 3.4836517031376743`*^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->{3.4694020398368797`*^9, 3.471554778448366*^9, 
  3.471577255865014*^9, 3.483651703173944*^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 = \\!\\(672\\), the memory in use = \
\\!\\(30417264\\), the time used = \\!\\(9.71733`\\)\"\>",
  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, 672, 30417264, 9.71733],
  
  Editable->False]], "Print",
 CellChangeTimes->{3.4694020398368797`*^9, 3.471554778448366*^9, 
  3.471577255865014*^9, 3.483651703176426*^9}],

Cell[BoxData[
 InterpretationBox["\<\"FindDeterminingEquations is done. The memory in use = \
\\!\\(30418008\\), the time used = \\!\\(9.717535999999999`\\)\"\>",
  StringForm[
  "FindDeterminingEquations is done. The memory in use = ``, the time used = \
``", 30418008, 9.717535999999999],
  Editable->False]], "Print",
 CellChangeTimes->{3.4694020398368797`*^9, 3.471554778448366*^9, 
  3.471577255865014*^9, 3.483651703178228*^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->{3.4694020398368797`*^9, 3.471554778448366*^9, 
  3.471577255865014*^9, 3.483651703179892*^9}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0.041799999999998505`", ",", "Null"}], "}"}]], "Output",
 CellChangeTimes->{3.469402039993959*^9, 3.4715547787535343`*^9, 
  3.47157725603944*^9, 3.4836517031823177`*^9}]
}, Open  ]],

Cell[CellGroupData[{

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

Cell[BoxData["\<\"D[y1[x1],x1,x1]-D[y1[x1],x1]-y1[x1]^3\"\>"], "Output",
 CellChangeTimes->{3.469402040077347*^9, 3.4715547788448763`*^9, 
  3.471577256121537*^9, 3.483651703224531*^9}]
}, Open  ]],

Cell[CellGroupData[{

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

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0", ",", 
   RowBox[{
    RowBox[{"-", "3"}], " ", 
    SuperscriptBox[
     RowBox[{"y1", "[", "x1", "]"}], "2"]}], ",", 
   RowBox[{"-", "1"}], ",", "1"}], "}"}]], "Output",
 CellChangeTimes->{3.469402040136787*^9, 3.471554778912437*^9, 
  3.4715772561809597`*^9, 3.483651703278557*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["The program expresses the determining equations in \
terms of zvariables. Here is the correspondence between z-variables and \
conventional variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData["ztableofrules"], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"z1", "\[Rule]", "x"}], ",", 
   RowBox[{"z2", "\[Rule]", 
    RowBox[{"y", "[", "x", "]"}]}]}], "}"}]], "Output",
 CellChangeTimes->{3.469402040202504*^9, 3.47155477897919*^9, 
  3.471577256248412*^9, 3.4836517033247557`*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Here are the determining equations expressed in terms \
of z-variables. This output may be quite long. The equations in the table can \
be distinguished by the ==0 at the end of each item.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData["zdeterminingequations"], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"\<\"-Derivative[0, 2][xse1][z1, z2] == 0\"\>", 
   ",", "\<\"-2*Derivative[0, 1][xse1][z1, z2] + Derivative[0, 2][eta1][z1, \
z2] - 2*Derivative[1, 1][xse1][z1, z2] == 0\"\>", 
   ",", "\<\"-3*z2^2*eta1[z1, z2] + z2^3*Derivative[0, 1][eta1][z1, z2] - \
Derivative[1, 0][eta1][z1, z2] - 2*z2^3*Derivative[1, 0][xse1][z1, z2] + \
Derivative[2, 0][eta1][z1, z2] == 0\"\>", 
   ",", "\<\"-3*z2^3*Derivative[0, 1][xse1][z1, z2] - Derivative[1, \
0][xse1][z1, z2] + 2*Derivative[1, 1][eta1][z1, z2] - Derivative[2, \
0][xse1][z1, z2] == 0\"\>"}], "}"}]], "Output",
 CellChangeTimes->{3.4694020402702513`*^9, 3.471554779043983*^9, 
  3.471577256314836*^9, 3.48365170338035*^9}]
}, Open  ]],

Cell["How many determining equations are there?", "Text"],

Cell[CellGroupData[{

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

Cell[BoxData["4"], "Output",
 CellChangeTimes->{3.469402040343314*^9, 3.471554779117277*^9, 
  3.47157725638548*^9, 3.483651703453536*^9}]
}, Open  ]],

Cell["\<\
Lets look at the determining equations in a slightly more readable form.\
\>", "Text",
 FontWeight->"Plain",
 FontVariations->{"CompatibilityType"->0}],

Cell[BoxData[
 RowBox[{"zdeterminingequationscolumn", "=", 
  TagBox[
   FormBox[
    RowBox[{"Column", "[", 
     RowBox[{
      RowBox[{"ToExpression", "[", "zdeterminingequations", "]"}], ",", 
      RowBox[{"{", 
       RowBox[{"Left", ",", "Below"}], "}"}]}], "]"}],
    TraditionalForm],
   TraditionalForm,
   Editable->True]}]], "Input"],

Cell[BoxData[
 TagBox[GridBox[{
    {
     RowBox[{
      RowBox[{"-", 
       RowBox[{
        SuperscriptBox["xse1", 
         TagBox[
          RowBox[{"(", 
           RowBox[{"0", ",", "2"}], ")"}],
          Derivative],
         MultilineFunction->None], "[", 
        RowBox[{"z1", ",", "z2"}], "]"}]}], "\[Equal]", "0"}]},
    {
     RowBox[{
      RowBox[{
       RowBox[{
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        RowBox[{
         SuperscriptBox["xse1", 
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           RowBox[{"(", 
            RowBox[{"0", ",", "1"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"z1", ",", "z2"}], "]"}]}], "+", 
       RowBox[{
        SuperscriptBox["eta1", 
         TagBox[
          RowBox[{"(", 
           RowBox[{"0", ",", "2"}], ")"}],
          Derivative],
         MultilineFunction->None], "[", 
        RowBox[{"z1", ",", "z2"}], "]"}], "-", 
       RowBox[{"2", " ", 
        RowBox[{
         SuperscriptBox["xse1", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"1", ",", "1"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"z1", ",", "z2"}], "]"}]}]}], "\[Equal]", "0"}]},
    {
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"-", "3"}], " ", 
        SuperscriptBox["z2", "2"], " ", 
        RowBox[{"eta1", "[", 
         RowBox[{"z1", ",", "z2"}], "]"}]}], "+", 
       RowBox[{
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        RowBox[{
         SuperscriptBox["eta1", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"0", ",", "1"}], ")"}],
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         RowBox[{"z1", ",", "z2"}], "]"}]}], "-", 
       RowBox[{
        SuperscriptBox["eta1", 
         TagBox[
          RowBox[{"(", 
           RowBox[{"1", ",", "0"}], ")"}],
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        RowBox[{"z1", ",", "z2"}], "]"}], "-", 
       RowBox[{"2", " ", 
        SuperscriptBox["z2", "3"], " ", 
        RowBox[{
         SuperscriptBox["xse1", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"1", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"z1", ",", "z2"}], "]"}]}], "+", 
       RowBox[{
        SuperscriptBox["eta1", 
         TagBox[
          RowBox[{"(", 
           RowBox[{"2", ",", "0"}], ")"}],
          Derivative],
         MultilineFunction->None], "[", 
        RowBox[{"z1", ",", "z2"}], "]"}]}], "\[Equal]", "0"}]},
    {
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"-", "3"}], " ", 
        SuperscriptBox["z2", "3"], " ", 
        RowBox[{
         SuperscriptBox["xse1", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"0", ",", "1"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"z1", ",", "z2"}], "]"}]}], "-", 
       RowBox[{
        SuperscriptBox["xse1", 
         TagBox[
          RowBox[{"(", 
           RowBox[{"1", ",", "0"}], ")"}],
          Derivative],
         MultilineFunction->None], "[", 
        RowBox[{"z1", ",", "z2"}], "]"}], "+", 
       RowBox[{"2", " ", 
        RowBox[{
         SuperscriptBox["eta1", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"1", ",", "1"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"z1", ",", "z2"}], "]"}]}], "-", 
       RowBox[{
        SuperscriptBox["xse1", 
         TagBox[
          RowBox[{"(", 
           RowBox[{"2", ",", "0"}], ")"}],
          Derivative],
         MultilineFunction->None], "[", 
        RowBox[{"z1", ",", "z2"}], "]"}]}], "\[Equal]", "0"}]}
   },
   GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Below}}},
   GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}],
  "Column"]], "Output",
 CellChangeTimes->{3.469402040403372*^9, 3.4715547791784267`*^9, 
  3.471577256448299*^9, 3.483651703525428*^9}],

Cell["\<\
Now solve the determining equations in terms of multivariable polynomials of \
a selected order. The Mathematica function Solve uses Gaussian elimination to \
solve a large number of linear equations for the polynomial coefficients. The \
time roughly follows

time/timeref=((number of equations)/(number of equationsref))^n

where the exponent is between 2.4 and 2.7. The Mathematica function Timing \
outputs the time required for the SolveDeterminingEquations function to \
execute.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Timing", "[", 
  RowBox[{"SolveDeterminingEquations", "[", 
   RowBox[{
   "independentvariables", ",", "dependentvariables", ",", "r", ",", "xseon", 
    ",", "zdeterminingequations", ",", 
    RowBox[{"order", "=", "4"}]}], "]"}], "]"}]], "Input"],

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 = \\!\\(15\\)\"\>",
  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 = ``", 15],
  Editable->False]], "Print",
 CellChangeTimes->{3.469402040478005*^9, 3.471554779318345*^9, 
  3.471577256519998*^9, 3.483651703591528*^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->{3.469402040478005*^9, 3.471554779318345*^9, 
  3.471577256519998*^9, 3.483651703625839*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(30458576\\), The time = \\!\
\\(9.737003`\\)\"\>",
  StringForm["The memory in use = ``, The time = ``", 30458576, 9.737003],
  Editable->False]], "Print",
 CellChangeTimes->{3.469402040478005*^9, 3.471554779318345*^9, 
  3.471577256519998*^9, 3.483651703668109*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The number of unknown polynomial coefficients = \
\\!\\(30\\)\"\>",
  StringForm["The number of unknown polynomial coefficients = ``", 30],
  Editable->False]], "Print",
 CellChangeTimes->{3.469402040478005*^9, 3.471554779318345*^9, 
  3.471577256519998*^9, 3.483651703697576*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The number of equations for the polynomial \
coefficients = \\!\\(51\\)\"\>",
  StringForm[
  "The number of equations for the polynomial coefficients = ``", 51],
  Editable->False]], "Print",
 CellChangeTimes->{3.469402040478005*^9, 3.471554779318345*^9, 
  3.471577256519998*^9, 3.483651703730555*^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->{3.469402040478005*^9, 3.471554779318345*^9, 
  3.471577256519998*^9, 3.483651703764063*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(30444392\\), The time = \\!\
\\(9.739008`\\)\"\>",
  StringForm["The memory in use = ``, The time = ``", 30444392, 9.739008],
  Editable->False]], "Print",
 CellChangeTimes->{3.469402040478005*^9, 3.471554779318345*^9, 
  3.471577256519998*^9, 3.483651703797776*^9}],

Cell[BoxData["\<\"Solve has finished.\"\>"], "Print",
 CellChangeTimes->{3.469402040478005*^9, 3.471554779318345*^9, 
  3.471577256519998*^9, 3.483651703830262*^9}],

Cell[BoxData["\<\"The function SolveDeterminingEquations is finished \
executing.\"\>"], "Print",
 CellChangeTimes->{3.469402040478005*^9, 3.471554779318345*^9, 
  3.471577256519998*^9, 3.483651703867981*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(30452680\\), The time = \\!\
\\(9.740041`\\)\"\>",
  StringForm["The memory in use = ``, The time = ``", 30452680, 9.740041],
  Editable->False]], "Print",
 CellChangeTimes->{3.469402040478005*^9, 3.471554779318345*^9, 
  3.471577256519998*^9, 3.483651703897162*^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->{3.469402040478005*^9, 3.471554779318345*^9, 
  3.471577256519998*^9, 3.4836517039316187`*^9}],

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