explicit versus implicit solution

When you finally pin down a solution, it can arrive in two different shapes. Sometimes you get y served up cleanly on its own, as a tidy formula: y equals some expression in x. Other times the best you can do is an equation that ties x and y together without isolating y — the relationship is genuine, but y is tangled up inside it. The first is explicit, the second implicit.

Precisely, an explicit solution gives the dependent variable directly as a function of the independent one, y = f(x), so you can read off y for any x by plugging in. An implicit solution is a relation F(x, y) = C that the solution satisfies but in which y has not been (or cannot be) solved for in elementary terms; the curve is defined by the equation rather than by an output formula. For example x^2 + y^2 = C is an implicit solution of y' = -x/y: differentiating x^2 + y^2 = C with respect to x gives 2x + 2y y' = 0, which rearranges to exactly y' = -x/y, so the relation holds even though y is not isolated.

Implicit solutions are perfectly legitimate and extremely common — for many equations, especially separable and exact ones, an implicit relation is simply the natural form the answer takes, and forcing it into explicit form may be impossible or may need a case-by-case choice of sign or branch. So the practical attitude is: an implicit solution is a complete answer. You verify it the same way, by implicit differentiation, and you only solve for y when you actually need explicit values, accepting that you may have to pick the right branch of the curve.