I'm not clear yet how to write the math symbols.
Therefore I'm wirting in plain text.
The separable differential equation I'm dealing with ends up as:
arctan(y) = arctan(x) + K
How do I find the solutions from here?
The possible problem I see here is that is going to return a reference angle, so technically your solution will be
where n is an integer.
I suppose the source (as it were) of the differential equation will have to tell you how to handle that.
I'm not sure if this answers your question....