Exact Equations

For the next technique it is best to consider first-order differential equations written in differential form

                                                         (1)

where and are given functions, assumed to be sufficiently smooth. The method that we will consider is based on the idea of a differential. Recall from a previous calculus course that if is a function of two variables, and , then the differential of , denoted , is defined by

                                                               (2)

EXAMPLE: Solve

                                                         (3)



In the foregoing example we were able to write the given differential equation in the form , and hence obtain its solution. However, we cannot always do this. Indeed we see by comparing equation (1) with (2) that the differential equation



can be written as if and only if



for some function . This motivates the following definition:

DEFINITION: The differential equation



is said to be exact in a region of the -plane if there exists a function such that

                                                            (4)

for all in Any function satisfying (4) is called a potential function for the differential equation



We emphasize that if such a function exists, then the preceding differential equation can be written as



This is why such a differential equation is called an exact differential equation. From the previous example, a potential function for the differential equation



is



We now show that if a differential equation is exact and we can find a potential function , its solution can be written down immediately.

THEOREM: The general solution to an exact equation



is defined implicitly by



where satisfies (4) and is an arbitrary constant.

THEOREM (Test for Exactness): Let , and their first partial derivatives and , be continuous in a (simply connected) region of the -plane. Then the differential equation



is exact for all in if and only if

                                                                       (5)

EXAMPLE: Determine whether the given differential equation is exact.

1.  

2.  

EXAMPLE: Find the general solution to




EXAMPLE: Find the general solution to





Integrating Factors


Usually a given differential equation will not be exact. However, sometimes it is possible to multiply the differential equation by a nonzero function to obtain an exact equation that can then be solved using the technique we have described in this section. Notice that the solution to the resulting exact equation will be the same as that of the original equation, since we multiply by a nonzero function.

DEFINITION: A nonzero function is called an integrating factor for the differential equation



if the differential equation



is exact.

EXAMPLE: Show that is an integrating factor for the differential equation

                                               (14)


THEOREM: The function is an integrating factor for

                                                       (20)

if and only if it is a solution to the partial differential equation

                                                  (21)

THEOREM: Consider the differential equation



1. There exists an integrating factor that is dependent only on if and only if



a function of only. In such a case, an integrating factor is



2. There exists an integrating factor that is dependent only on if and only if



a function of only. In such a case, an integrating factor is



EXAMPLE: Solve

                                                  (22)