Finding the characteristic function of a 2nd order ODE with variables on both sides

Question

The given function is $$x^2y^{''}+xy^{"}+y=\ln(x)$$ Now finding the characteristic or aux. function is very simple, usually. This problem has $x$ terms on both sides of the equation. Would it be better to move all the $x$ terms to the right hand side, thus making the equation: $$y^{"}+y^{'}+y=\frac{\ln(x)}{x^3}$$ Or should it be done after the change of the $y$ terms, the reasoning being that the equation Aux. now will have a variable within the $m$ term: $$m^2x^2+mx+1=0$$ $$(m+1)(x^2m+x)=0$$ $$m_1=-1 \space m_2=-\frac{1}{x}$$ But I do see this being a far more complex way to solve, if not flat out wrong. Should I try to remove the $x$ terms before the change to $m$'s occurs or should I solve $m$ in terms of $x$?

Answer 1

$$x^2y''+xy'+y=\ln(x)$$ Substitute $x=e^t$ the equation becomes: $$y''+y=t$$ Which is easier to integrate.