## Exact First-Order Differential Equations

The differential equation

F(x,y) dx + G(x,y) dy = 0

is exact if
∂G/∂x
= ∂F/∂y in some simply
connected region of the plane. Solution to an exact
differential equation may be attained by determing f(x,y) where f satisfies:

∇f(x,y)
= (F(x,y),
G(x,y))

Example:

Example:

(y

^{2}- 2x) dx + (2xy + 1) dy = 0Determine if the above differential equation is exact. If so, solve the differential equation.

Solution: