Use the package IntroToSymmetry.m to work out the point group of the second order ODE

Yxx+Yx^2/Y-1/Y=0 

considered in homework exercise 8.9.

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, ie, in quotes.

inputequation=
"D[y[x],x,x]+D[y[x],x]^2/y[x]-1/y[x]";

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.

rulesarray=
{"D[y[x],x,x]->-D[y[x],x]^2/y[x]+1/y[x]"};

Enter the list of independent variables.

independentvariables={"x"};

Enter the list of dependent variables.

dependentvariables={"y"};

Enter the list of function names and constants that need to be protected when the equation is converted to generic (x1,y1) variables.

frozennames={};

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 one sets 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.

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

equationgenericvariables

invarconditiontable

Look at the determining equations

zdeterminingequations

Here are the determining equations in a slightly more readable form as expressions rather than strings.

zdeteqns=ToExpression[zdeterminingequations]

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

ztableofrules

Now solve the determining equations in terms of multivariable polynomials of a selected order.

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 input variables.

ToExpression[xsefunctions]/.ztableofrules

Express the eta function in terms of these variables.

ToExpression[etafunctions]/.ztableofrules

Here is a list of the various infinitesimal groups.

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

Column[infinitesimalgroupsxy]

Check that these symmetries satisfy the determining equations.

Simplify[ToExpression[zdeterminingequations]]

The final result indicates that the equation is invariant under three groups. The parameter a10 corresponds to a translation in x, the parameter b14 represents a dilation in x and y and b15 is similar to a projective group. Now generate the commutator table of the Lie algebra. First the number of groups

MakeCommutatorTable[independentvariables,dependentvariables,infinitesimalgroupsxy]

Put the commutator table in readable form.

MatrixForm[commutatortable]

This is a nonsolvable Lie algebra. Two of the groups can be selected to produce a two-dimensional that Lie algebra that can be used to reduce the equation to quadrature.

MaxMemoryUsed[]