Conditions to apply variation of constant méthod on a Cauchy problem

Question

I try to solve the following Cauchy Problem on the interval $[0,T[$ for a $T>0$:

$ \left\{\begin{matrix}y'(t) = -\frac{4}{t} y(t) \\ y(0)=0 \end{matrix}\right. $

What I did:

I found the variation of constant method which gives the unique solution $e^{\int_{0}^{t}\frac{-4}{s}ds}y(0)=0$ . But I am not convinced that $t \rightarrow \frac{-4}{t}$ is continuous on [0,T] to ensure that the integral have a meaning.

Many thanks if you can point me to the right direction.

Answer 1

note that $ -\int_{0}^{t}\frac{4}{s}ds=\int_{t}^{0}\frac{4}{s}ds=\lim_{x \rightarrow 0}(4\text{log}(x))-4\text{log}(t)$, where the limit is $-\infty$. Now if some $f \rightarrow -\infty$ , what happens to $e^{f}$ ?