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

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?



ode:=x^2*diff(y(x),x)^2-(1+2*x*y(x))*diff(y(x),x)+1+y(x)^2 = 0;

x^2*(diff(y(x), x))^2-(1+2*x*y(x))*(diff(y(x), x))+1+y(x)^2 = 0


[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

Vector(3, {(1) = y(x) = (1/4)*(4*x^2-1)/x, (2) = y(x) = _C1*x-sqrt(_C1-1), (3) = y(x) = _C1*x+sqrt(_C1-1)})


`casesplit/ans`([(diff(y(x), x))^2 = (2*y(x)*(diff(y(x), x))*x+diff(y(x), x)-y(x)^2-1)/x^2], [2*(diff(y(x), x))*x^2-2*x*y(x)-1 <> 0]), `casesplit/ans`([y(x) = (1/4)*(4*x^2-1)/x], [])

ode:=convert(ode,D): #solve for y(x) first, this will generate 2 ODE's

Vector(2, {(1) = y(x) = (diff(y(x), x))*x+sqrt(diff(y(x), x)-1), (2) = y(x) = (diff(y(x), x))*x-sqrt(diff(y(x), x)-1)})


[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

y(x) = _C1*x+(_C1-1)^(1/2)

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

y(x) = _C1*x-(_C1-1)^(1/2)


`casesplit/ans`([diff(y(x), x)-1 = (-(diff(y(x), x)-1)^(1/2)+y(x)-x)/x, diff(y(x), x) = (-(diff(y(x), x)-1)^(1/2)+y(x))/x], [1+2*x*(diff(y(x), x)-1)^(1/2) <> 0]), `casesplit/ans`([y(x) = (1/4)*(4*x^2-1)/x], [])


`casesplit/ans`([diff(y(x), x)-1 = ((diff(y(x), x)-1)^(1/2)+y(x)-x)/x, diff(y(x), x) = ((diff(y(x), x)-1)^(1/2)+y(x))/x], [2*x*(diff(y(x), x)-1)^(1/2)-1 <> 0]), `casesplit/ans`([y(x) = (1/4)*(4*x^2-1)/x], [])




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