Pensieve header: Ultra-fine oscillation criteria.

eq = y''[x]+γ y[x] /. y -> (Y[(E^(#))]&) /.
x -> Log[z];
eq= Expand[(eq/(Coefficient[eq, Y''[z]]))]

{p,q} = Coefficient[eq, #]& /@
{Y'[z], Y[z]};
Q=q-(1/4)(p^2)-(1/2)(∂_z)p // Simplify

Eq0 = (x^2)y''[x]+γ y[x]

Eq1=Eq0 /. y -> (Y[(E^(#))]&) /. x -> Log[z]

Eq2= Expand[(Eq1/(Coefficient[Eq1, Y''[z]]))]

{p,q} = Coefficient[Eq2, #]& /@ {Y'[z], Y[z]}

Q=q-(1/4)(p^2)-(1/2)(∂_z)p

Eq3 = v''[z]+Q v[z]

Eq3 /. {v->y, z->x}

UltraFine[0] = (x^2)y''[x]+γ y[x];
UltraFine[n_] /; n>0 := (
Eq0=UltraFine[n-1];
Eq1=Eq0 /. y -> (Y[(E^(#))]&) /. x -> Log[z];
Eq2= Expand[(Eq1/(Coefficient[Eq1, Y''[z]]))];
{p,q} = Coefficient[Eq2, #]& /@ {Y'[z], Y[z]};
Q=q-(1/4)(p^2)-(1/2)(∂_z)p;
Eq3 = v''[z]+Q v[z];
Eq3 /. {v->y, z->x}
);

UltraFine[1]

UltraFine[2]

UltraFine[3]