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

and *N* integrated with respect to *y*:

(with each “constant” of integration ignored, as usual). These calculations clearly give

as the general solution of the differential equation.

**Example 2:** The equation

is not exact, since

However, note that

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

the given differential equation will have

as an integrating factor. Multiplying the differential equation through by μ = (sin *y*) ^{−1} yields

which *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

is the general solution of the differential equation.

**Example 3:** Solve the IVP

The given differential equation is not exact, since

However, note that

which can be interpreted to be, say, a function of *x* only; that is, this last equation can be written as ξ( *x*) ≡ 2. Case 1 then says that

will be an integrating factor. Multiplying both sides of the differential equation by μ( *x*) = *e* ^{2 x }yields

which *is* exact because

Now, since

and

(with the “constant” of integration suppressed in each calculation), the general solution of the differential equation is

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

Thus, the particular solution is

which can be expressed explicitly as

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

has an integrating factor of the form μ( *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,

is exact for some *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 M *dx* + N *dy* = 0 reads

Now, since

and

(ignoring the “constant” of integration in each case), the general solution of the differential equation (*)—and hence the original differential equation—is clearly