Explain briefly the use of the method of characteristics to solve linear first-order partial differential equations.

Use the method to solve the problem

$$(x-y) \frac{\partial u}{\partial x}+(x+y) \frac{\partial u}{\partial y}=\alpha u$$

where $\alpha$ is a constant, with initial condition $u(x, 0)=x^{2}, x \geqslant 0$.

By considering your solution explain:

(i) why initial conditions cannot be specified on the whole $x$-axis;

(ii) why a single-valued solution in the entire plane is not possible if $\alpha \neq 2$.