Question

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.

Answer 1

The solution to your linear differential equation is $\phi(x) = x^c F(x)$ where $F(x)$ is any antiderivative of $f(x) x^{-c}$. In order for some solution to have $\lim_{x \to 0} x^{-c} \phi(x)$ exist, the necessary and sufficient condition is that the improper integral $\int_0^\varepsilon f(x)\ x^{-c}\ dx$ converges for some $\varepsilon > 0$. Then the limit will exist for all the solutions. However, the limit will be $0$ for one of those solutions, namely $\phi(x) = x^c \int_0^x f(t)\ t^{-c}\ dt$.