How to find an integrating factor for this problem?

It would take a pretty lucky guess to come up with an integrating factor directly. However, you can use a Riccati transform and then back-track to find an integrating factor if that's your pleasure.

If you substitute y(x) = - x (dw/dx) / (9 w) where w is a function of x, you'll wind up with a linear 2nd order ODE in w.

Wikipedia has a decent description: http://en.wikipedia.org/wiki/Riccati_equation

 

I sure don't see an obvious way. Is it possible that the equation should be

y dy - (y + x^2 + 9 y^2) dy = 0 ?

An integrating factor would be simple to find in that case.

 

group in terms of 3y/x to make the integrating factor easy to see
x dy - (y + x^2 + 9y^2)dx = 0
(x²/3)d(3y/x)-x²((3y/x)²+1)dx=0

 

It helps that I knew that
x dy-y dx=x²d(y/x)
from there it was clear x and y/x would be better variables that x and y.
These take a bit of practice.
There are a few others, but I keep in mind these common examples
d(xy)=x dy+y dx
y² d(x/y)=y dx-x dy
x² d(y/x)=x dy-y dx
(x²+y²) d(arctan(y/x))=x dy-y dx
[tex] x^{1-p}y^{1-q}{d}(x^py^q)=pydx+qxdy[/tex]