I am reading a proof of the measurable mapping theorem which shows the existence of a solution to the Beltrami equation in the simple case when $\mu \in L^{\infty}(\mathbb{C})$ is real analytic (due to Gauss)
\begin{equation} (1-\mu(x,y))\frac{\partial{f}}{\partial{x}}+ i(1+\mu(x,y)))\frac{\partial{f}}{\partial{y}}=0 \tag{4.5.8} \end{equation}
From what I understand, we are looking for a curve $(x(t), y(t))$ which satisfies
$$\frac{dx}{dt}=1-\mu(x,y)$$ and $$\frac{dy}{dt}=i(1+\mu(x,y))$$.
The author then points out that such a curve would satisfy $$\frac{dy}{dx}= i\frac{1+u}{1-u} \tag{*}$$
I wanted to understand how one goes from the system of ODE's above to (*). Does $\mu$ being real analytic imply that there is a solution to the system above
Furthermore, if there is such a solution, the inverse function theorem tells us that the solution is a local homeomorphism, can we say anything about uniqueness just from the ODE theory?