Let $f,g:\mathbb{R}\rightarrow \mathbb{R}$ functions such that $g$ is continuous and $f$ is Lipschitz. Consider the IVP
$$ \frac{dx}{dt}=\sigma(t,x)\quad x(t_0)=x_0$$
where $\sigma:\mathbb{R}\times\mathbb{R}^2\rightarrow \mathbb{R}^2$ is defined by $$ \sigma(t,x)=(f(x_1),g(x_1)x_2)$$
for all $t\in \mathbb{R}$ and $x=(x_1,x_2)\in \mathbb{R}^2$. Prove that the given IVP has unique solution in every interval (where it is defined). ¿Can we take off the assumption that $f$ is Lipschitz and obtain the same conclusion?
Can somebody give me a hint whit this problem?
The above equation is essentially $2$ equations - \begin{align*} \frac{dx_1}{dt} &= f(x_1) \\ \frac{dx_2}{dt} &= g(x_1)x_2 \end{align*} As $f(x)$ is Lipschitz, the $1st$ IVP has a unique solution $x_1(t)$, which is differentiable. Thus, as $g(x)$ is continuous, we get $g(x_1(t))$ as integrable.
Hence, for the $2nd$, we have \begin{align*} \frac{dx_2}{dt} &= g(x_1)x_2 \\ \implies \int_{x_{0_2}}^{x_2} \frac{dx_2}{x_2}&= \int_{t_0}^t g(x_1(t))dt \\ \implies x_2 &= x_{0_2} e^{\int_{t_0}^t g(x_1(t))dt} \end{align*}