Exit

Block[{Print},<<xAct`xTras`]

$DefInfoQ=False;

DefManifold[M,6,{a,b,c,d,e,g,h,i,j,k,l,m,n,o,p}]

DefMetric[-1,metric[-c,-d],CD,PrintAs->"g",CurvatureRelations->True]

PrintAs[RiemannCD]^="R";
PrintAs[RicciCD]^="R";
PrintAs[RicciScalarCD]^="R";
PrintAs[ChristoffelCD]^="Γ";
PrintAs[EinsteinCD]^="G";

DefScalarFunction [F];

DefTensor[Fp[],M,PrintAs->SubscriptBox["F","R"]]

F/:Perturbation[F[RicciScalarCD[]]]:=Fp[]Perturbation[RicciScalarCD[]]

abseom=FullSimplification[]@VarL[metric[a,b],CD][F[RicciScalarCD[]]]

DefChart[sp,M,{0,1,2,3,4,5},{t[],x1[],x2[],x3[],y[],θ[]}]

DefScalarFunction[{A,B}]

bg=CTensor[DiagonalMatrix[{A[y[]]^2,A[y[]]^2,A[y[]]^2,A[y[]]^2,1,B[y[]]^2}],{-sp,-sp}];
invbg=Inv[bg];
cd=LC[bg];
ricciscalar=RicciScalar[cd];
riccicd=Ricci[cd];

DefScalarFunction[f]
DefScalarFunction[fR,PrintAs->SubscriptBox["f","R"]]

yrule={
F[RicciScalarCD[]]->f[y[]],
Fp[]->fR[y[]]
};

rule1={
metric[a_?DownIndexQ,b_?DownIndexQ]:>bg[a,b],
metric[a_?UpIndexQ,b_?UpIndexQ]:>invbg[a,b],
CD[a_]:>cd[a]
};

rule2={
RicciScalarCD[]:>ricciscalar[],
RicciCD[a_,b_]:>riccicd[a,b]
};

eom=FromBasisExpand[abseom//.rule1]/.yrule/.rule2//Simplify

collect[expr_]:=Collect[ContractBasis[expr],{fR[y[]],fR'[y[]],fR''[y[]],fR'''[y[]]},Expand]

equ1=collect[-eom Basis[{1,-sp},a]Basis[{1,-sp},b]/A[y[]]^2]

equ4=collect[-eom Basis[{4,-sp},a]Basis[{4,-sp},b]]

equ5=collect[-eom Basis[{5,-sp},a]Basis[{5,-sp},b]/B[y[]]^2]