First, you need to realize that, just as an equation may not be exact, it may also not have an integrating factor that is a pure function of just x or y. Say you start with:
M(x,y)dx + N(x,y)dy = 0
Of course your first test would be to see if N_{x} = M_{y}. If so, it is exact and you proceed with exact methods. But what if it isn't exact. Maybe, if we are lucky, we can find an integrating factor of the form μ(x) or μ(y) that will make it exact. So let's try it:
μ(x)M(x,y)dx + μ(x)N(x,y)dy = 0
Let's try the exactness test on this. We need
(μ(x)N(x,y))_{x} = (μ(x)M(x,y))_{y}
μ'(x)N(x,y) + μ(x)N_{x}(x,y) = μ(x)M_{y}(x,y)
[tex]\mu'(x) = \frac {\mu(x)(M_y(x,y)  N_x(x,y))}{N(x,y)}[/tex]
We can only hope to find μ(x) as a pure function of x if there are no y's on the right hand side. So before we even try to find such a μ(x), we should test the given equation to see if
[tex]\frac {M_y(x,y)  N_x(x,y)}{N(x,y)}[/tex]
is a pure function of x.
Trying the same thing for a pure function of y we get:
μ(y)M(x,y)dx + μ(y)N(x,y)dy = 0
Testing for exactness:
μ(y)N_{x}(x,y) = μ(y)M_{y}(x,y) + μ'(y)M(x,y)
[tex]\mu'(y) = \frac {\mu(y)(N_x(x,y)  M_y(x,y))}{M(x,y)}[/tex]
For this to work there needs to be no x on the right side, so:
[tex]\frac {M_y(x,y)  N_x(x,y)}{M(x,y)}[/tex]
must be a pure function of y.
In your example, neither of the two tests work, which explains why you didn't get a pure function of y to integrate and your method failed.
