Use the package DeterminingEquations.m to work out the point group of of the 2rd order ODE

Yxx-Exp[y]/(xYx)=0.

First read in the package

Needs["SymmetryAnalysis`IntroToSymmetry`"]

Enter the input equation as a string, ie, in quotes.

inputequation=
"D[y[x],x,x]-Exp[y[x]]/(x*D[y[x],x])";

The function y[x] is a solution of the equation and this constraint must be applied to the invariance condition. Replace the second derivativewherever it appears in the invariance condition.

rulesarray=
{"D[y[x],x,x]->Exp[y[x]]/(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/or constant names that must be preserved when the equation is expressed in terms of generic (x1,y1) variables.

frozennames = {"Exp"};

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-Bäcklund groups. For the usual case where one is searching for point groups one sets r=0.

r = 0;

When searching for Lie-Bäcklund 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 and in string format.

zdeterminingequations

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

zdeteqns=ToExpression[zdeterminingequations]

We need to remove the x factors which appear in the denominator of the determining equations. First remove the ==  0 from each determining equation string. This is a string operation so be careful to include the space between the second equals sign and the zero.

zdeterminingequationsnoequalszero=
StringReplace[zdeterminingequations,
{"== 0"->" "}];

Now convert each determining equation to an expression and multiply through by z1^2. The built-in function Together cancels terms in the numerator and denominator.

zdeterminingequationsnoequalszeroexpression=
Together[(z1^2)*ToExpression[
zdeterminingequationsnoequalszero]];

The number of determining equations.

imax=Length[
zdeterminingequationsnoequalszeroexpression]

Convert each determining equation back to a string and reinstate the == 0.

zdeterminingequationsmod=Table[
ToString[InputForm[
zdeterminingequationsnoequalszeroexpression[[i]]]]<>
"== 0",
{i,1,imax}];

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

ztableofrules

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

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

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,zdeterminingequationsmod,order=4]]

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

Here is a list of the various infinitesimal groups.

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

ColumnForm[infinitesimalgroupsxy]

This process has correctly captured the group (xse,eta)=(-x/2,1).

ColumnForm[infinitesimalgroupsxy]

The presence of Exp[y] in the determining equations suggests that we search for trancendental solutions of the determining equations. To make further progress let's use the following form of the infinitesimals to try to simplify the determining equations.

xse1[z1_, z2_] := -b10*(z1/2) + f[z1, z2]

eta1[z1_, z2_] := b10 + g[z1, z2]

Substitute these functions into the expression form of the determining equations.

zdeterminingequationscolumn

Simplify[%]

Encouraged by the 4th and 5th equations in the last step, we now try the ansatz,

xse1[z1_, z2_] := -b10*(z1/2) + f[z1]

eta1[z1_, z2_] := b10 + g[z2]

Substitute these functions into the expression form of the determining equations.

zdeterminingequationscolumn

Simplify[%]

More effort is required here but so far it looks like the the only solution is the result given by the package.

MaxMemoryUsed[ ]