potential function
When an exact equation hides a single underlying function whose level curves are the solutions, that function has a name: the potential function, usually written F. The word 'potential' is borrowed from physics, where the gravitational or electrostatic potential is a height-like quantity whose slopes give the force. Here F plays the same role — its slopes (partial derivatives) are exactly the coefficients M and N of the equation.
Formally, F(x, y) is a potential for M dx + N dy = 0 when partial F / partial x = M and partial F / partial y = N. Then M dx + N dy is just dF, and the implicit solution is F(x, y) = C. Geometrically the solution curves are the contour lines of the surface z = F(x, y): along each contour the height does not change, which is precisely dF = 0.
The potential function is what makes exact equations special and easy: instead of integrating a differential equation, you reconstruct one function and read its level sets. The same idea reappears later as conserved quantities for planar systems and as Hamiltonian or gradient structure — in each case a hidden scalar function organizes the whole flow.
For (3x^2 + y) dx + (x - 2y) dy = 0 the potential is F(x, y) = x^3 + xy - y^2, since partial F / partial x = 3x^2 + y = M and partial F / partial y = x - 2y = N. The solutions are the curves x^3 + xy - y^2 = C.
The potential F turns a differential equation into a family of contour lines F = C.
A potential function is unique only up to an added constant: if F works, so does F + 5, since the extra constant gets absorbed into C. So never worry about a stray constant when reconstructing F.