An exact differential equation has the following form

,      (1)

where the partial derivatives of  P(x,y) and  Q(x,y) obey the condition

.           (2)

Due to condition (2) and in view of the theorem of a total differential, the expression on the left-hand side of equation (1) is the total differential of some function  u(x,y):

.

Therefore,

,      (3)

,     (4)

and

If we hold  y  fixed, then by integrating equation (3) with respect to x, we obtain

.            (5)

Note that constant of integration depends on the variable y because y is considered as a constant in the integral with respect to x.

To find , substitute expression (5)  into equation (4):

.      (6)

This is an ordinary differential equation for the function .

Since  depends on the variable y only, the expression on the right-hand side of (4) must also be a function of y only. Otherwise, equation (1) is not an exact differential equation.

By solving equation (6), we find a particular solution    and hence, the general integral

u(x,y) = C.

Indefinite Integrals Definite Integrals Differential Equations