The Crime Scene: Dividing by Zero
In the previous guide you learned to solve a separable equation by splitting the variables: take dy/dx = f(x) g(y), move all the y's to one side and all the x's to the other, then integrate. The very first move is to divide both sides by g(y). That single division is innocent-looking, but it carries a hidden assumption — that g(y) is not zero. Wherever g(y) does equal zero, you have just divided by zero, and the method silently walks past those values without ever recording them.
Take the friendly example dy/dx = y. Here g(y) = y, so the forbidden value is y = 0. Dividing by y gives (1/y) dy = dx, and integrating yields ln|y| = x + C, then y = A e^x. That family looks complete — but notice it can never produce y = 0, because A e^x is zero only if A = 0, and A = e^C is strictly positive (a constant of exponentiation can't be zero). Yet plug y = 0 straight into the original equation: dy/dx = 0 and y = 0, so 0 = 0. It checks out perfectly. The constant function y = 0 is a genuine solution that our tidy formula threw in the bin.
What Got Lost Were the Flat Lines
Here is the beautiful part: the lost values are not strange exotic curves. When g(y0) = 0, the constant function y(x) = y0 makes dy/dx = 0 on the left and f(x) g(y0) = 0 on the right — both sides zero, for every x. So y = y0 is a flat, horizontal solution that sits there forever. These are precisely the equilibrium solutions you met earlier: the heights at which the slope field is dead level and a solution, once placed, never moves. Separation of variables, by its nature, can only ever find the *moving* solutions — the ones where y actually changes — so the resting ones slip through.
Geometrically the picture is vivid. The slope field is carved into horizontal strips by these flat lines, and the curves the formula produces live inside the strips, pressing up against the boundaries but never crossing them. The boundary lines themselves — the equilibria — are solutions too, but they belong to a different category. That is exactly why dividing by g(y) erased them: you can't slide along a line of zero slope using a method built to track changing slopes.
dy/dx = y (1 - y) [the logistic equation]
g(y) = y(1 - y) = 0 -> y = 0 and y = 1
y = 1 ___________________ <- equilibrium (lost)
\ moving solutions live here
\_/\___ curves climb from 0 toward 1
y = 0 ___________________ <- equilibrium (lost)Sometimes the Loss Actually Hurts
You might shrug: who cares about a single flat line? Often it is harmless — if your initial condition starts you strictly inside a strip, you stay there, and the missing equilibrium never affects your answer. But sometimes the lost line is the whole story. Consider the logistic equation dy/dx = y(1 - y) modeling a population. If you carelessly write the general solution as y = 1/(1 + A e^(-x)) and forget y = 0, you've lost the answer to a perfectly real question: a population that starts at zero stays at zero. Extinction is an equilibrium, and your formula can't see it.
There is a deeper warning hiding here too. A lost constant solution is closely tied to the failure of uniqueness. The notorious example y' = y^(2/3) with y(0) = 0 has BOTH the equilibrium y = 0 AND the moving solution y = (x/3)^3 passing through the same starting point — two different solutions from one initial condition. The flat line you'd otherwise drop is one of the competing answers. Throw it away and you might never notice that the problem doesn't even have a single, well-defined solution. That is a much bigger deal than a missing tickmark on a graph.
Are Lost Solutions the Same as Singular Solutions?
It's worth being precise, because two ideas brush against each other here. The constant solution you recover by checking g(y) = 0 usually turns out to be a member of the general family in disguise — often it's the limiting case A -> infinity or A = 0 of your formula, just unreachable by any finite constant. But there is a rarer, stranger animal: a true singular solution, one that is NOT contained in the general solution for any value of the arbitrary constant at all, even in a limit. These often appear as an envelope that the whole family of curves hugs without belonging to it.
The practical upshot is the same either way: always check the values where g(y) = 0 by hand. Whether the flat line turns out to be a hidden limit of the family or a genuinely independent singular solution, you only ever discover it by substituting it back into the original equation — which is exactly the habit of verifying by substitution you've been building all along. The formula tells you what it can compute; substitution tells you the truth.
The Fix: A Three-Beat Habit
The cure costs almost nothing once it becomes reflex. Before you ever divide, pause and read off the danger points; integrate as usual; then circle back and audit. Done this way, separation of variables loses nothing — you simply restore by hand the solutions the algebra had to skip. Here is the habit, beat by beat.
- Before dividing, solve g(y) = 0. Each root y0 gives a constant candidate y = y0 — write them down as your list of possibly-lost equilibrium solutions.
- Now divide by g(y) and integrate normally to get the moving family (the general solution with its arbitrary constant).
- Check each constant candidate: is y = y0 already produced by the family for some value of the constant (possibly a limit)? If yes, it's covered. If no, it is a separate solution you must add back explicitly.
- Finally, if you have an initial condition, decide which solution actually passes through it — and watch for the case where more than one does, the red flag of non-uniqueness.