I don't think can be counted as a complete answer, as I left out all the mathematical details to the OP.
I missed the original question, answers, as well as this discussion. I just discover them now, after the request for reopening vote.
I have read more than once all related material and I think this is a good case to discuss a little more in general, with respect to this specific case, how the site policy about homework-like question is implemented. Notice that I am not discussing the policy itself. Just its application.
The first observation is that it is not clear to me why this question was originally classified as homework-like. It is true that the form of the request was apparently stressing more the mathematical side of the problem and that the OP wrote that he/she tried to obtain a function relation without succeeding, thus exposing the question to the possible classification of homework.
However, I think that in a case like the present, the border between homework-like and conceptual question is not well definite. And even the border between Physics and Math is not always clear. Physics SE contains many accepted questions and answers on QM or GR topics which could easily be classified as pure math.
Let me explain better my point of view.
The original question was analyzing a particular case, by trying to obtain an answer by brute force. That is something in the direction of a homework-like question. However, it is not difficult to see how it could be recast in the form of a perfectly admissible conceptual question for the Physics site. It would be enough to stress i) the interest in Physics for obtaining a phase space portrait of classical motions and ii) the fact that the case of the damped harmonic oscillator is an example of the more general question about the conditions on the motion which allow to get a functional dependence of the velocity on the position.
Was this the intention of the OP? We cannot know. Maybe it was a real homework or the OP has not enough experience to drive the attention to the underlying conceptual problem. But even if the original question was really originating from a homework (please, take onto account that the tag homework was not put by the OP), still I do not see why people answering to the question could not have been able to provide a conceptual, general answer. By the way, this could be the simplest method to avoid providing a complete answer to the specific problem. In this respect, I think that a behavior more consistent with the reasons for our policy against home-work like questions, in a borderline case like the present, would be not to close the question if there is any chance that it could trigger good conceptual answers. I would rather check the kind and quality of the answers.