Let $g,f : \mathbb{R} \rightarrow \mathbb{R}$ be continuous, $f$ being lipschitzian. Show that the system
$\left\{\begin{array}{l} x'=f(x), & x(t_0)=x_0\\ y'=g(x)y, & y(t_0)=y_0 \end{array} \right.$
has a unique solution on any interval (where it is defined). Can one eliminate the hypothesis that $f$ is Lipschitzian and obtain the same conclusion?