In this example we use the package IntroToSymmetry.m to work out the groups for the third-order ODE considered in Homework Exercise 8.11.

Yxxx+(Yx/Y)Yxx-(2/Yx)Yxx^2+(Yx/Y^2)=0

Clear all symbols in the current context.

ClearAll[Evaluate[Context[]<>"*"]]

First read in the package which is located in User Home Folder/Library/Mathematica/Applications/SymmetryAnalysis.

Needs["SymmetryAnalysis`IntroToSymmetry`"]

Enter the input equation as a string. Don't include the ==0 at the end.

inputequation=
"D[y[x],x,x,x]+(D[y[x],x]/y[x])*D[y[x],x,x]-(2/D[y[x],x])*(D[y[x],x,x])^2+D[y[x],x]/(y[x])^2";

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. Replace the highest derivative term.

rulesarray=
{"D[y[x],x,x,x]->-((D[y[x],x]/y[x])*D[y[x],x,x]-(2/D[y[x],x])*(D[y[x],x,x])^2+D[y[x],x]/(y[x])^2)"};

Enter the list of independent variables.

independentvariables={"x"};

Enter the list of dependent variables.

dependentvariables={"y"};

Enter the list of function and constant names that need to be preserved when the variables are changed to generic names, ( x1,y1).

frozennames={""};

Enter the maximum derivative order of the equation.

p=3;

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.

r=0;

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.

xseon=1;

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.

internalrules=0;

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.

Timing[FindDeterminingEquations[
independentvariables,dependentvariables,frozennames,p,r,xseon,
inputequation,rulesarray,internalrules]]

equationgenericvariables

invarconditiontable

Here are the determining equations in terms of z-variables.

zdeterminingequations

ztableofrules

ToExpression[zdeterminingequations]/.ztableofrules

Here is the correspondence between z-variables and conventional variables.

ztableofrules

Lets look at the determining equations in a slightly more readable form.

zdeterminingequationscolumn=Column[ToExpression[zdeterminingequations],{Left,Below}]

Now solve the determining equations in terms of multivariable polynomials of a selected order. This function can take a long time. 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.

Timing[SolveDeterminingEquations[
independentvariables,dependentvariables,r,
xseon,zdeterminingequations,order=3]]

Here are the infinitesimal transformation functions for the independent variables.

xsefunctions

and for the dependent variables.

etafunctions

Express the xse functions in terms of x,y variables.

ToExpression[xsefunctions]/.ztableofrules

Express the eta function in terms of these variables.

ToExpression[etafunctions]/.ztableofrules

infinitesimalgroupsxy=infinitesimalgroups/.{z1->x,z2->y}

Here is a list of the various infinitesimal groups.

Column[infinitesimalgroupsxy]

Check that these symmetries satisfy the determining equations.

Simplify[ToExpression[zdeterminingequations]]

The package finds two groups, the translation group (xse,eta)=(1,0) and the dilation group (xse,eta)=(x,y). Let's generate the commutator table of the group.

MakeCommutatorTable[
independentvariables,dependentvariables,infinitesimalgroupsxy]

MatrixForm[commutatortable]

MaxMemoryUsed[ ]