In this example we use the package IntroToSymmetry.m to work out the groups for several second-order ODEs listed in Table 8.1.

First read in the package which is located in a folder called SymmetryAnalysis inside the Mathematica folder.

Needs["SymmetryAnalysis`IntroToSymmetry`"]

Enter the input equation as a string. Don't include the == 0 at the end. The function f[Y,Yx]is arbitrary.

\(\(inputequation = 
"D[y[x],x,x]-f[x,D[y[x],x]]";\)\)

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.

\(\(rulesarray = 
{"D[y[x],x,x]->f[x,D[y[x],x]]"};\)\)

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 equation is converted to generic (x1,y1) variables.

frozennames = {"f"};

Enter the maximum derivative order of the equation.

p = 2;

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

ztableofrules

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.

zdeterminingequations

How many determining equations are there?

Length[zdeterminingequations]

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

zdeterminingequationscolumn = 
      TraditionalForm[
        ColumnForm[ToExpression[zdeterminingequations], Left, 
          Below]]

Note that a number of the determining equations are redundant.

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.

Timing[
      SolveDeterminingEquations[
independentvariables, 
        dependentvariables, r, 
xseon, 
        zdeterminingequations, order = 2]]

Here are the infinitesimals for the independent variables expressed in terms of z-variables.

xsefunctions

and the infinitesimals for the dependent variables.

etafunctions

The package finds the invariance under translation in Y.