I am not sure if this is correct, but this would be my thinking.

When we want to solve a second order linear DE, the solution is the sum of the the homogeneous and the non-homogeneous solutions.

Since we can write this DE as , I would expect you could use a similar process here, where the solution would be equal to the sum of the solutions of (homogeneous case), (first order linear case) and (Bernoulli case).

Is there anything wrong with my thinking?