(* Content-type: application/mathematica *)

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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-a*Y^(-3)=0 \n\n\
considered in homework excercise 8.5.",
 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[
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  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}, {3.472046901826345*^9, 
  3.472046902384349*^9}},
 ImageRegion->{{0, 1}, {0, 1}}],

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

Cell[TextData[StyleBox["Enter the input equation as a string. Don't include \
the == 0 at the end. The constant a  is arbitrary.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"inputequation", "=", "\n", "\"\<D[y[x],x,x]-a*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]->a*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 equation is converted to (x1,y1) variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"frozennames", "=", 
   RowBox[{"{", "\"\<a\>\"", "}"}]}], ";"}]], "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 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. 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[{
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     "p", ",", "r", ",", "xseon", ",", "\[IndentingNewLine]", "inputequation",
     ",", "rulesarray", ",", "internalrules"}], "]"}], "]"}]], "Input"],

Cell[BoxData[
 InterpretationBox["\<\"The function FindDetermining Equations has begun, the \
memory in use = \\!\\(26168072\\), the time used = \\!\\(1.013028`\\)\"\>",
  StringForm[
  "The function FindDetermining Equations has begun, the memory in use = ``, \
the time used = ``", 26168072, 1.013028],
  Editable->False]], "Print",
 CellChangeTimes->{3.4693981575247307`*^9, 3.471552111811718*^9, 
  3.47157681299793*^9, 3.483553673440372*^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->{3.4693981575247307`*^9, 3.471552111811718*^9, 
  3.47157681299793*^9, 3.483553673443014*^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 = \\!\\(600\\), the memory in use = \
\\!\\(26165136\\), the time used = \\!\\(1.04142`\\)\"\>",
  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, 600, 26165136, 1.04142],
  
  Editable->False]], "Print",
 CellChangeTimes->{3.4693981575247307`*^9, 3.471552111811718*^9, 
  3.47157681299793*^9, 3.4835536734456043`*^9}],

Cell[BoxData[
 InterpretationBox["\<\"FindDeterminingEquations is done. The memory in use = \
\\!\\(26165880\\), the time used = \\!\\(1.041616`\\)\"\>",
  StringForm[
  "FindDeterminingEquations is done. The memory in use = ``, the time used = \
``", 26165880, 1.041616],
  Editable->False]], "Print",
 CellChangeTimes->{3.4693981575247307`*^9, 3.471552111811718*^9, 
  3.47157681299793*^9, 3.4835536734476633`*^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->{3.4693981575247307`*^9, 3.471552111811718*^9, 
  3.47157681299793*^9, 3.483553673449461*^9}],

Cell[BoxData[
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  RowBox[{"0.029214000000000073`", ",", "Null"}], "}"}]], "Output",
 CellChangeTimes->{3.46939815753659*^9, 3.4715521119423103`*^9, 
  3.471576813204541*^9, 3.483553673452444*^9}]
}, Open  ]],

Cell[CellGroupData[{

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

Cell[BoxData["\<\"D[y1[x1],x1,x1]-a*y1[x1]^(-3)\"\>"], "Output",
 CellChangeTimes->{3.469398157591198*^9, 3.471552112011917*^9, 
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}, Open  ]],

Cell[CellGroupData[{

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

Cell[BoxData[
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  RowBox[{"0", ",", 
   FractionBox[
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    SuperscriptBox[
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}, Open  ]],

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

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"z1", "\[Rule]", "x"}], ",", 
   RowBox[{"z2", "\[Rule]", 
    RowBox[{"y", "[", "x", "]"}]}]}], "}"}]], "Output",
 CellChangeTimes->{3.469398157754507*^9, 3.4715521121455183`*^9, 
  3.4715768134598837`*^9, 3.483553673598621*^9}],

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[{"\<\"-(z2^4*Derivative[0, 2][xse1][z1, z2]) == 0\"\>", 
   ",", "\<\"z2^4*Derivative[0, 2][eta1][z1, z2] - 2*z2^4*Derivative[1, \
1][xse1][z1, z2] == 0\"\>", 
   ",", "\<\"3*a*eta1[z1, z2] + a*z2*Derivative[0, 1][eta1][z1, z2] - \
2*a*z2*Derivative[1, 0][xse1][z1, z2] + z2^4*Derivative[2, 0][eta1][z1, z2] \
== 0\"\>", 
   ",", "\<\"-3*a*z2*Derivative[0, 1][xse1][z1, z2] + 2*z2^4*Derivative[1, \
1][eta1][z1, z2] - z2^4*Derivative[2, 0][xse1][z1, z2] == 0\"\>"}], 
  "}"}]], "Output",
 CellChangeTimes->{3.4693981578097258`*^9, 3.4715521122132473`*^9, 
  3.471576813563611*^9, 3.483553673649722*^9}]
}, Open  ]],

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

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

Cell[BoxData[
 TagBox[GridBox[{
    {
     RowBox[{
      RowBox[{
       RowBox[{"-", 
        SuperscriptBox["z2", "4"]}], " ", 
       RowBox[{
        SuperscriptBox["xse1", 
         TagBox[
          RowBox[{"(", 
           RowBox[{"0", ",", "2"}], ")"}],
          Derivative],
         MultilineFunction->None], "[", 
        RowBox[{"z1", ",", "z2"}], "]"}]}], "\[Equal]", "0"}]},
    {
     RowBox[{
      RowBox[{
       RowBox[{
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         RowBox[{"z1", ",", "z2"}], "]"}]}]}], "\[Equal]", "0"}]},
    {
     RowBox[{
      RowBox[{
       RowBox[{"3", " ", "a", " ", 
        RowBox[{"eta1", "[", 
         RowBox[{"z1", ",", "z2"}], "]"}]}], "+", 
       RowBox[{"a", " ", "z2", " ", 
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         SuperscriptBox["eta1", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"0", ",", "1"}], ")"}],
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         RowBox[{"z1", ",", "z2"}], "]"}]}], "-", 
       RowBox[{"2", " ", "a", " ", "z2", " ", 
        RowBox[{
         SuperscriptBox["xse1", 
          TagBox[
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            RowBox[{"1", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"z1", ",", "z2"}], "]"}]}], "+", 
       RowBox[{
        SuperscriptBox["z2", "4"], " ", 
        RowBox[{
         SuperscriptBox["eta1", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"2", ",", "0"}], ")"}],
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          MultilineFunction->None], "[", 
         RowBox[{"z1", ",", "z2"}], "]"}]}]}], "\[Equal]", "0"}]},
    {
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"-", "3"}], " ", "a", " ", "z2", " ", 
        RowBox[{
         SuperscriptBox["xse1", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"0", ",", "1"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"z1", ",", "z2"}], "]"}]}], "+", 
       RowBox[{"2", " ", 
        SuperscriptBox["z2", "4"], " ", 
        RowBox[{
         SuperscriptBox["eta1", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"1", ",", "1"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"z1", ",", "z2"}], "]"}]}], "-", 
       RowBox[{
        SuperscriptBox["z2", "4"], " ", 
        RowBox[{
         SuperscriptBox["xse1", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"2", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"z1", ",", "z2"}], "]"}]}]}], "\[Equal]", "0"}]}
   },
   GridBoxAlignment->{"Columns" -> {{Left}}},
   GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}],
  "Column"]], "Output",
 CellChangeTimes->{3.4693981578735743`*^9, 3.4715521122788153`*^9, 
  3.471576813663589*^9, 3.4835536737010803`*^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", ",", 
    "\[IndentingNewLine]", "xseon", ",", "zdeterminingequations", ",", 
    RowBox[{"order", "=", "4"}]}], "]"}], "]"}]], "Input",
 CellChangeTimes->{{3.4693981448900557`*^9, 3.469398145535972*^9}}],

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->{3.469398157925312*^9, 3.471552112350533*^9, 
  3.471576813886339*^9, 3.483553673750745*^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->{3.469398157925312*^9, 3.471552112350533*^9, 
  3.471576813886339*^9, 3.483553673753566*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(26211736\\), The time = \\!\
\\(1.05732`\\)\"\>",
  StringForm["The memory in use = ``, The time = ``", 26211736, 1.05732],
  Editable->False]], "Print",
 CellChangeTimes->{3.469398157925312*^9, 3.471552112350533*^9, 
  3.471576813886339*^9, 3.483553673754993*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The number of unknown polynomial coefficients = \
\\!\\(30\\)\"\>",
  StringForm["The number of unknown polynomial coefficients = ``", 30],
  Editable->False]], "Print",
 CellChangeTimes->{3.469398157925312*^9, 3.471552112350533*^9, 
  3.471576813886339*^9, 3.4835536737562*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The number of equations for the polynomial \
coefficients = \\!\\(47\\)\"\>",
  StringForm[
  "The number of equations for the polynomial coefficients = ``", 47],
  Editable->False]], "Print",
 CellChangeTimes->{3.469398157925312*^9, 3.471552112350533*^9, 
  3.471576813886339*^9, 3.483553673757762*^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->{3.469398157925312*^9, 3.471552112350533*^9, 
  3.471576813886339*^9, 3.483553673759098*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(26221752\\), The time = \\!\
\\(1.059326`\\)\"\>",
  StringForm["The memory in use = ``, The time = ``", 26221752, 1.059326],
  Editable->False]], "Print",
 CellChangeTimes->{3.469398157925312*^9, 3.471552112350533*^9, 
  3.471576813886339*^9, 3.483553673760277*^9}],

Cell[BoxData["\<\"Solve has finished.\"\>"], "Print",
 CellChangeTimes->{3.469398157925312*^9, 3.471552112350533*^9, 
  3.471576813886339*^9, 3.483553673761417*^9}],

Cell[BoxData["\<\"The function SolveDeterminingEquations is finished \
executing.\"\>"], "Print",
 CellChangeTimes->{3.469398157925312*^9, 3.471552112350533*^9, 
  3.471576813886339*^9, 3.483553673762724*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(26232904\\), The time = \\!\
\\(1.061507`\\)\"\>",
  StringForm["The memory in use = ``, The time = ``", 26232904, 1.061507],
  Editable->False]], "Print",
 CellChangeTimes->{3.469398157925312*^9, 3.471552112350533*^9, 
  3.471576813886339*^9, 3.483553673763892*^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->{3.469398157925312*^9, 3.471552112350533*^9, 
  3.471576813886339*^9, 3.483553673765852*^9}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0.014804999999999957`", ",", "Null"}], "}"}]], "Output",
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  3.471576814276635*^9, 3.483553673766945*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Here are the infinitesimals for the independent \
variables expressed in terms of z-variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData["xsefunctions"], "Input"],

Cell[BoxData[
 RowBox[{"{", "\<\"xse1[z1_, z2_]=a10 + 2*b15*z1 + b16*z1^2\"\>", 
  "}"}]], "Output",
 CellChangeTimes->{3.469398157992488*^9, 3.4715521128804407`*^9, 
  3.471576814330682*^9, 3.483553673816437*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["and the infinitesimals for the dependent variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData["etafunctions"], "Input"],

Cell[BoxData[
 RowBox[{"{", "\<\"eta1[z1_, z2_]=b15*z2 + b16*z1*z2\"\>", "}"}]], "Output",
 CellChangeTimes->{3.469398158059464*^9, 3.471552112947027*^9, 
  3.471576814431122*^9, 3.48355367386621*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Express the xse functions in terms of x,y variables.",
 
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[CellGroupData[{

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