We show how multiplying an equation by an integrating factor can make the equation exact, and we give examples where this is a nice technique for solving a first-order equation.

### Integrating Factors

In Trench 2.5, Exact Equations, we saw that if , , and are continuous and on an open rectangle then

is exact on . Sometimes an equation that isn’t exact can be made exact by multiplying it by an appropriate function. For example, is not exact, since in (eq:2.6.2). However, multiplying (eq:2.6.2) by yields which is exact, since in (eq:2.6.3). Solving (eq:2.6.3) by Procedure proc:solvingExactEq of Trench 2.5, yields the implicit solutionA function is an *integrating factor* for (eq:2.6.1) if

#### Finding Integrating Factors

By applying Theorem thmtype:2.5.2 (with and replaced by and ), we see that (eq:2.6.4) is exact on an open rectangle if , , , and are continuous and on . It’s better to rewrite the last equation as

which reduces to the known result for exact equations; that is, if then (eq:2.6.5) holds with , so (eq:2.6.1) is exact.You may think (eq:2.6.5) is of little value, since it involves *partial* derivatives of the
unknown integrating factor , and we haven’t studied methods for solving such
equations. However, we’ll now show that (eq:2.6.5) is useful if we restrict our search to
integrating factors that are products of a function of and a function of ; that is,
. We’re not saying that *every* equation has an integrating factor of this
form; rather, we’re saying that *some* equations have such integrating factors.
We’ll now develop a way to determine whether a given equation has such an
integrating factor, and a method for finding the integrating factor in this
case.

If , then and , so (eq:2.6.5) becomes

or, after dividing through by , Now let so (eq:2.6.7) becomesWe obtained (eq:2.6.8) by *assuming* that has an integrating factor . However, we can now
view (eq:2.6.7) differently: If there are functions and that satisfy (eq:2.6.8) and we define

There’s no simple general method for ascertaining whether functions and satisfying (eq:2.6.8) exist. However, the next theorem gives simple sufficient conditions for the given equation to have an integrating factor that depends on only one of the independent variables and , and for finding an integrating factor in this case.

- (a)
- If is independent of on and we define then is an integrating factor for on
- (b)
- If is independent of on and we define then is an integrating factor for (eq:2.6.11) on

- Proof
- item:2.6.1a If is independent of , then (eq:2.6.8) holds with and . Therefore
so (eq:2.6.10) is an integrating factor for (eq:2.6.11) on .
item:2.6.1b If is independent of then eqrefeq:2.6.8 holds with and , and a similar argument shows that (eq:2.6.12) is an integrating factor for (eq:2.6.11) on .

The next two examples show how to apply Theorem thmtype:2.6.1.

To solve this equation, we must find a function such that

and Integrating (eq:2.6.16) with respect to yields Differentiating this with respect to yields Comparing this with (eq:2.6.15) shows that ; therefore, we can let in (eq:2.6.17) and conclude that is an implicit solution of (eq:2.6.14). It is also an implicit solution of (eq:2.6.13).The figure below shows a direction field and some integral curves for (eq:2.6.13)

Theorem thmtype:2.6.1 does not apply in the next example, but the more general argument that led to Theorem thmtype:2.6.1 provides an integrating factor.

The figure below shows a direction field and integral curves for (eq:2.6.23).

In (eq:2.6.26) and We look for functions and such that that is,

The right side will contain the term if . Then (eq:2.6.28) becomes so . Since and we can take and , which yields the integrating factor . Multiplying (eq:2.6.26) by yields the exact equation We leave it to you to show that this equation has the implicit solution Solving for yields which we rewrite as by renaming the arbitrary constant. This is also a solution of (eq:2.6.26).The figure below shows a direction field and some integral curves for (eq:2.6.26).

### Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)