If asked to find the limiting water level I could set dy/dt to zero and solve for y, then verify that values above and below that y are positive/negative respectively.

I have no attempt at solving that DE, no methods gone over thus far in the book seem applicable. This is in Apostol Calculus Vol I., section 8.28 #19. For first-order differential equations he covered homogeneous and separable options, as well as Bernoulli's equation. If there is a way to solve this, and my way is indeed cheating, please let me know because I do want to do this properly.

I know, in theory, that I would solve the DE for y in terms of t, then take the limit as t increases without bound.

WolframAlpha's solution to the DE does not seem to illuminate any solution to the problem, however.