If a differential equation of the form

*x,y*) such that the equivalent equation obtained by multiplying both sides of (*) by μ,

*is* exact. Such a function μ is called an **integrating factor** of the original equation and is guaranteed to exist if the given differential equation actually has a solution. *Integrating factors turn nonexact equations into exact ones*. The question is, how do you find an integrating factor? Two special cases will be considered.

*Case 1:*

Consider the differential equation

M dx+N dy= 0. If this equation is not exact, thenM_{y }will not equalN_{x }; that is,M_{y }–N_{x}≠ 0. However, if

is a function of xonly, let it be denoted by ξ(x). Then

will be an integrating factor of the given differential equation.

*Case 2:*

Consider the differential equation

M dx+N dy= 0. If this equation is not exact, thenM_{y}will not equalN_{x }; that is,M_{y}–N_{x }≠ 0. = 0. However, if

is a function of yonly, let it be denoted by ψ(y). Then

will be an integrating factor of the given differential equation.

**Example 1:** The equation

is not exact, since

*x* alone. Therefore, by Case 1,

*x* yields

*is* exact because

Solving this equivalent exact equation by the method described in the previous section, *M* is integrated with respect to *x*,

*N* integrated with respect to *y*:

**Example 2:** The equation

However, note that

*y* alone (Case 2). Denote this function by ψ( *y*); since

*y*) ^{−1} yields

*is* exact because

To solve this exact equation, integrate *M* with respect to *x* and integrate *N* with respect to *y*, ignoring the “constant” of integration in each case:

These integrations imply that

**Example 3:** Solve the IVP

However, note that

*x* only; that is, this last equation can be written as ξ( *x*) ≡ 2. Case 1 then says that

*x*) = *e* ^{2 x }yields

*is* exact because

Now, since

The value of the constant *c* is now determined by applying the initial condition *y*(0) = 1:

Thus, the particular solution is

**Example 4:** Given that the nonexact differential equation

*x,y*) = *x* ^{a }*y* ^{b }for some positive integers *a* and *b*, find the general solution of the equation.

Since there exist positive integers *a* and *b* such that *x* ^{a }*y* ^{b }is an integrating factor, multiplying the differential equation through by this expression must yield an exact equation. That is,

*a* and *b*. Exactness of this equation means

By equating like terms in this last equation, it must be the case that

The simultaneous solution of these equations is *a* = 3 and *b* = 1.

Thus the integrating factor *x* ^{a }*y* ^{b }is *x* ^{3} *y*, and the exact equation *dx* + *dy* = 0 reads

Now, since