Infolevel confirms that dsolve integrates a first order ode
intde := int(x^2*diff(y(x), x)/(x^2 - 1), x) = int(y(x)^(1/2), x)^(-2/3);
infolevel[dsolve] := 5;
dsolve(intde);
Methods for first order ODEs:
--- Trying classification methods ---
trying homogeneous types:
differential order: 1; looking for linear symmetries
trying exact
<- exact successful
If you additionally increase printlevel up to 50 you will find the ODE (which can also be derived by differention of the original equation w.r.t. to x).
Edit: In this way Maple reduces the inital equation to a first oder equation.
Then appropriate methods are applied
This is the first order integroDE Maple integrates. An ODE without integral that integrates to the same result would be this one
To get there, you have to find a way to convert the right handside of your problem like this (i.e. getting the integration out of the denominator):
The attachment shows where the above right hand side comes from.
integrode_backward.mw