a lost constant solution
Here is a small trap that catches almost everyone once. When you separate variables you divide by g(y). But division is only legal when g(y) is not zero — and any value of y that makes g(y) = 0 gives a perfectly good constant solution that your formula quietly drops. A lost constant solution is exactly one of these vanished answers.
Picture the equation dy/dx = y(1 - y). To separate, you divide by y(1 - y), which silently assumes y is neither 0 nor 1. But the constant functions y = 0 and y = 1 BOTH satisfy the original equation (each makes the right side zero, so dy/dx = 0, which a constant indeed has). They are equilibrium solutions, and they are not produced by any choice of the constant in your integrated family — they were thrown out the moment you divided. The fix is a habit: before dividing by g(y), find every root of g(y) = 0, check that the corresponding constant is a genuine solution, and append it to your answer list.
This matters because those missing solutions are often the most important ones physically: they are the steady states a system can rest at. In a logistic population model, y = 0 (extinction) and y = carrying capacity are precisely the lost constants, and an answer that omits them is incomplete. The deeper lesson is that an algebraic manipulation can change the solution set, so methods that divide must be audited for what they discard.
Solving dy/dx = y by separation gives y = A e^x. With A = 0 you do get y = 0, so here nothing is truly lost. But for dy/dx = y^2 - y you divide by y^2 - y and must restore y = 0 and y = 1 by hand — neither comes from the integrated family.
Sometimes the constant solution is recovered by a special parameter value; sometimes it is genuinely outside the family and must be added back.
Whether y = 0 is 'lost' depends on the equation: in dy/dx = y it is just A = 0, but in dy/dx = y^(1/2) it is a separate solution the family cannot reach. Always check the roots of g(y) explicitly.