exact equation
Imagine you are standing on a hillside and someone hands you tiny instructions: 'if you step a little east, your height changes by this much; if you step a little north, it changes by that much.' If those two instructions are consistent — if they really came from one underlying landscape — then there is a hidden height function, and you can reconstruct the contour lines you are walking along. An exact equation is exactly this situation for a differential equation: the equation is secretly telling you to stay on a level curve of some hidden function.
Concretely, a first-order equation written in the form M(x, y) dx + N(x, y) dy = 0 is called exact when its left side is the total differential of a single function F(x, y) — that is, when dF = M dx + N dy. This requires that the partial derivative of F with respect to x is M and the partial derivative of F with respect to y is N. When that holds, the whole equation just says dF = 0, so F(x, y) = C is the implicit solution: the solution curves are the level curves of F.
Why care? Because once you recognise an equation as exact, solving it is almost mechanical — no separating, no integrating factor, no guessing. The catch is that most equations are NOT exact as written. So the real skill is a quick test for exactness, plus a trick (an integrating factor) to repair an equation that just barely misses being exact.
Take (2xy) dx + (x^2 + 1) dy = 0. Here M = 2xy and N = x^2 + 1. The hidden function turns out to be F(x, y) = x^2 y + y, because partial F / partial x = 2xy = M and partial F / partial y = x^2 + 1 = N. So the solution is x^2 y + y = C, i.e. y = C / (x^2 + 1).
An exact equation collapses to dF = 0, so the answer is the level curves F = C.
Exactness depends on HOW the equation is written: multiplying both sides by a function can make an exact equation non-exact, or vice versa. That freedom is exactly what an integrating factor exploits.