Solving partial differential equation with dimensional analysis

Question

I have the partial differential equation to solve $$-y''+(y')^2-2y'/x=((e^y)/x^2)(\partial f/\partial z +a x),$$where the prime is $\partial/\partial x$ and $y$ is a function of $x$ and $z$. It is known that $[\partial f/\partial z]=x^0$ where $[]$ means dimensions relative to $x$. Also, $[a]=1/x$ where $a$ is a constant. It follows that the $\partial f/\partial z$ term does not contain any explicit dependence on $x$ and can then be treated as a constant in the integration with respect to $x$. An analytic solution can then be obtained. But am I missing something by only using the fact that the dimensionality with respect to $x$ is $x^0$ to declare the partial $f$ term to not depend on $x$ in the integration with respect to $x$?