According to many sources and Maple itself, d'Alembert ode has the form
$$ y(x) = x f(y'(x)) + g( y'(x) ) $$
https://www.maplesoft.com/support/help/maple/view.aspx?path=odeadvisor%2FdAlembert
This is a generalization of Clairaut's ODE $ y(x)= x y'(x)+ g(y'(x))$
http://mathworld.wolfram.com/ClairautsDifferentialEquation.html
Maple has a function which gives the type of an ODE. My question is, why does it say the following ODE's are d'Alembert? I do not see how these fit the pattern shown above.
First example
ode:=diff(y(x),x)=y(x)/x+1/x*sqrt(y(x)*x);
DEtools:-odeadvisor(ode)
[[_homogeneous, `class A`], _rational, _dAlembert]
$$ {\frac {\rm d}{{\rm d}x}}y \left( x \right) ={\frac {y \left( x \right) }{x}}+{\frac {\sqrt {y \left( x \right) x}}{x}} $$
Second example
ode:=diff(y(x),x)=exp(y(x)/x)+y(x)/x;
DEtools:-odeadvisor(ode);
[[_homogeneous, `class A`], _dAlembert]
$$ {\frac {\rm d}{{\rm d}x}}y \left( x \right) ={{\rm e}^{{\frac {y \left( x \right) }{x}}}}+{\frac {y \left( x \right) }{x}} $$
Third example
ode:=diff(y(x),x)=2*x*y(x)/(x^2-y(x)^2);
DEtools:-odeadvisor(ode);
[[_homogeneous, `class A`], _rational, _dAlembert]
$$ {\frac {\rm d}{{\rm d}x}}y \left( x \right) =2\,{\frac {y \left( x \right) x}{{x}^{2}- \left( y \left( x \right) \right) ^{2}}} $$
If these are really dAlembert, then what is the f(y') and g(y') in these ODE? I am not sure how to transform these ODE's to the general form of dAlembert in order to verify.
Your first DE can be written as $$y = x\; \frac{2 y' + 1 \pm \sqrt{1+4y'}}{2}$$ your second as $$ y = x \left( y' - W(\exp(y')\right) $$ (where $W$ is the Lambert W function), and your third as $$ y = x \;\frac{-1 \pm \sqrt{(y')^2+1}}{y'} $$