Consider a firstorder ODE in the slightly different form

(1) 
Such an equation is said to be exact if

(2) 
This statement is equivalent to the requirement that a Conservative Field exists, so that a scalar potential can be
defined. For an exact equation, the solution is

(3) 
where is a constant.
A firstorder ODE (1) is said to be inexact if

(4) 
For a nonexact equation, the solution may be obtained by defining an Integrating Factor of (6) so that the
new equation

(5) 
satisfies

(6) 
or, written out explicitly,

(7) 
This transforms the nonexact equation into an exact one. Solving (7) for gives

(8) 
Therefore, if a function satisfying (8) can be found, then writing
in equation (5) then gives

(11) 
which is then an exact ODE. Special cases in which can be found include dependent, dependent, and
dependent integrating factors.
Given an inexact firstorder ODE, we can also look for an Integrating Factor so that

(12) 
For the equation to be exact in and , the equation for a firstorder nonexact ODE

(13) 
becomes

(14) 
Solving for
gives

(15) 
which will be integrable if

(16) 
in which case

(17) 
so that the equation is integrable

(18) 
and the equation

(19) 
with known is now exact and can be solved as an exact ODE.
Given in an exact firstorder ODE, look for an Integrating Factor
. Then

(20) 

(21) 
Combining these two,

(22) 
For the equation to be exact in and , the equation for a firstorder nonexact ODE

(23) 
becomes

(24) 
Therefore,

(25) 
Define a new variable

(26) 
then
, so

(27) 
Now, if

(28) 
then

(29) 
so that

(30) 
and the equation

(31) 
is now exact and can be solved as an exact ODE.
Given an inexact firstorder ODE, assume there exists an integrating factor

(32) 
so
. For the equation to be exact in and , equation (7) becomes

(33) 
Now, if

(34) 
then

(35) 
so that

(36) 
and the equation

(37) 
is now exact and can be solved as an exact ODE.
Given a firstorder ODE of the form

(38) 
define

(39) 
Then the solution is

(40) 
If

(41) 
where

(42) 
then letting

(43) 
gives

(44) 

(45) 
This can be integrated by quadratures, so

(46) 

(47) 
References
Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed.
New York: Wiley, 1986.
© 19969 Eric W. Weisstein
19990526