I'm trying to solve the following problem:
Consider the family of Cauchy problems $$y'=\frac{1}{1+ty},~~t>0$$ with initial conditions $$y(0)=1+\frac{1}{n}.$$ I want to show that for every $n$ there exists a solution $y_n$ defined on the whole set $\{t \in \mathbb{R}: t \geq 0 \} $.
I'm stucked because I don't see how I could even apply Picard-Lindelöf theorem, since $f(t,y) =\frac{1}{1+ty}$ diverges on the subset $\{(t, y): y=-\frac{1}{t}\} \subseteq \mathbb{R}^2.$
What am I missing here? Any hint or suggestion is appreciated.