Let $u_{0}: \mathbb{R} \rightarrow \mathbb{R}, u_{0} \in C^{1}(\mathbb{R}), u_{0}(x) \geqslant 0$ for all $x \in \mathbb{R}$. Consider the partial differential equation for $u=u(x, y)$,

$$4 y u_{x}+3 u_{y}=u^{2}, \quad(x, y) \in \mathbb{R}^{2}$$

subject to the Cauchy condition $u(x, 0)=u_{0}(x)$.

i) Compute the solution of the Cauchy problem by the method of characteristics.

ii) Prove that the domain of definition of the solution contains

$$(x, y) \in \mathbb{R} \times\left(-\infty, \frac{3}{\sup _{x \in \mathbb{R}}\left(u_{0}(x)\right)}\right)$$