This might be an easy question in the theory of ordinary differential equation. But since I know very little about it, I posted it here and hope to get some answers or references.

Consider the equation $$x\phi'(x)-c\phi(x)=xf(x), x>0$$ where $c$ is some constant.

My question is that under what (growth) conditions, the solution $\phi(x)$ has the following property: the limit $$\lim_{x\to 0}x^{-c}\phi(x)$$ exists and nonzero?

Thank you very much for your help and sorry if this question is not appropriate here.