This is taken from the book of Bender and Orszag, problem 3.44.
Find the leading behavior as $x\rightarrow+\infty$ of the differential equation:
$x^3y'' - (2x^3 -x^2)y' +(x^3-x^2-1)y=0$
Explain the appearance of a logarithm in the leading behavior. It appears that $e^x$ is a solution and I'm not sure where the logarithm comes from. Usually, one uses the ansatz $y\sim e^{S(x)}$ and does a dominant balance but clearly this method cannot produce a logarithm.