Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function and $g:\mathbb{R}\rightarrow\mathbb{R}$ be a Lipschitz function. Would you help me to prove that the system of differential equation
$ x'=g(x)$ $y'=f(x)y $
with initial value $x(t_0)=x_0$ and $y(t_0)=y_0$ has a unique solution.
Could I prove the uniqueness solution of $x'=g(x)$, $x(t_0)=x_0$ by Gronwall Inequality first then use the result to prove the second?