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?
Best Answer
Your reasonning is correct. Since $g$ is Lipschitz and the first equation of the system involves only $x$, there is a unique solution $x(t)$ such that $x(t_0)=x_0$.
The second equation becomes $$ y'=f(x(t))\,y,\quad y(t_0)=y_0. $$ It is a linear equation and has a unique solution, given by $$ y(t)=y_0e\,^{\int_{t_0}^t f(x(s))ds}. $$