I noticed that in the Ricatti ODE introduction in this video (https://www.youtube.com/watch?v=-SHQ1FMyqIo), the form of the general solution is stated as y = y_1 + u and the Ricatti ODE becomes a Bernoulli ODE, which eventually becomes linear, while in this video (https://www.youtube.com/watch?v=3heIoKDhN3E), the stated general solution is y = y_1 + 1/v, and the Ricatti ODE becomes a linear ODE directly. Is this difference in the assumed general solution form the reason for the presence or absence of the intermediate Bernoulli ODE step? If not, why is the more complex y = y_1 + 1/v sometimes preferred, and what conditions determine whether the Ricatti ODE will become Bernoulli as an intermediate step, or will be linear immediately?
Also, what is the difference in a nutshell between the Ricatti ODE and the generalized Ricatti ODE?
Also, what is the difference in a nutshell between the Ricatti ODE and the generalized Ricatti ODE?