@nm You wrote: "but it is d'Alembert ode. Maple itself says so".
Yes, I see that Maple says so. But look up the definition of the d'Alembert ODE in the help page that I noted in my answer. Compare that with your ODE. Is yours of that form? No! To put it in that form, you need to square the two sides of your ODE as I explained before. Probably that's what Maple does (unwisely) when it declares that your ODE is of d'Alembert type.
If you replace the right-hand side of your ODE by its negative, you obtain a completely different ODE. However, the distinction between the two ODEs disappears when you square the two sides. You can have no assurance that the solution of the squared ODE is that of one or the other ODE.
It's not difficult to verify that the two solution produced by Maple's d'Alembert's method are solutions to the ODE with the changed sign. Neither one is a solution of your original ODE.