Exact Differential Equation:

A first order differential equation

 
 where M and N are functions of x and y, is called exact if there exists some function ƒ (x, y) such that

 
 In that case ƒ (x, y) = c is the basic solution of the provided differential equation.

 
 is an exact differential equation, since

 
Theorem: The enough and necessary condition for the equation Mdx + Ndy = 0 to be exact is

 
Proof: (Necessary Condition)

(Sufficient Condition)

 

Let U = ∫M dx, where y is saved as a constant in M.                              (i)

 
 Where F(y) = ∫ƒ(y) dy.                                (iv)