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

Yxx+(2Yx+2cYx^(3/2)+2Yx^2)/(x-Y)=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]+2*(D[y[x],x]/(x-y[x]))+2*c*(D[y[x],x]^(3/2)/(x-y[x]))+2*(D[y[x],x]^2/(x-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]->-
(2*(D[y[x],x]/(x-y[x]))+
2*c*(D[y[x],x]^(3/2)/(x-y[x]))+
2*(D[y[x],x]^2/(x-y[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 must be preserved when the equation is expressed in terms of generic (x1,y1) variables.

frozennames = {"c"};

Enter the maximum derivative order of the equation.

p = 2;

The maximum derivative order that the infinitesimalsare 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. 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 the form of strings.

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-y) 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 (x-y)^2. The built-in function Together cancels terms in the numerator and denominator.

zdeterminingequationsnoequalszeroexpression=
Together[((z1-z2)^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

Now solve the determining equations in terms of multivariable polynomials of some 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=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

Here is a list of the various infinitesimal groups.

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

ColumnForm[infinitesimalgroupsxy]

The final result indicates that the equation is invariant under three groups. The parameter b17 corresponds to a projective group, the parameter b14 represents a dilation in x and y and b10 is a translation in x and y. Now generate the commutator table.

MakeCommutatorTable[independentvariables, dependentvariables, 
      infinitesimalgroupsxy]

Put the commutator table in readable form.

MatrixForm[commutatortable]

This is a nonsolvable Lie algebra.