Question:why Maple sometimes shows singular solution to Clairaut and sometimes not?

Question:why Maple sometimes shows singular solution to Clairaut and sometimes not?

Maple 2019

I do not understand why Maple sometimes shows singular solution to Clairaut ODE and sometimes not.

Clairaut ODE has the form y(x) = x y'(x) + G(x, y')

In the following ODE when I ask Maple to dsolve it as is, it does give singular solution. Next, when solving explicity for y(x) first, which will generate 2 ODE's, each is Clairaut ODE, then ask Maple to dsolve each, now Maple no longer gives the singular solution. But when I solve each one of these ODE's, I see that there is the singular solution there. It must be there, since this is Clairaut ODE and it has singular solution.

When I do PDEtools:-casesplit on each of the two ODE's generated by solving for y(x) first, I see the singular solution there.

The question is, why Maple dsolve does not show the singular solution in the second case? And how to make it show it? Or did I do something wrong?

 > restart;
 > Typesetting:-Settings(typesetprime=true):
 > ode:=x^2*diff(y(x),x)^2-(1+2*x*y(x))*diff(y(x),x)+1+y(x)^2 = 0;

 > Vector([dsolve(ode,y(x))]); #now it shows singular solution (first one below)

 > PDEtools:-casesplit(ode)

 > ode:=convert(ode,D): #solve for y(x) first, this will generate 2 ODE's sol:=[solve(ode,y(x))]: odes:=Vector(map(z->y(x)=z,convert(sol,diff)))

 > dsolve(odes[1],y(x)); #where is singular solution?

 > dsolve(odes[2],y(x)); #where is singular solution?

 > PDEtools:-casesplit(odes[1])

 > PDEtools:-casesplit(odes[2])

 >