I am trying to solve the partial differential equation $x\ u_ x - u\ u_y = y$ with the initial condition $u(1,y) = y$ , using the mathod of characteristics. My problem is with y and z , I mean
$$\frac{dy}{dt} = -z$$ $$\frac{dz}{dt} = y$$ How can I solve these two? Is the solution unique? What is the maximal domain where it is defined?
On
Hint.
Regarding the PDE (it remembers a Hopf type PDE)
$$ x u_x-u u_y = y $$
we have
$$ \dot x = x\\ \dot y = -u\\ \dot u = y $$
from then we can extract
$$ y\dot y + u\dot u = 0\\ u\dot x+x\dot y = 0 $$
representing the characteristic families
$$ y^2+u^2 = C_1\\ $$
and after substitution
$$ \pm\frac{\dot y}{\sqrt{C_1-y^2}} =-\frac{\dot x}{x} $$
or
$$ \pm\arctan\left(\frac{y}{\sqrt{C_1-y^2}}\right) = -\ln x + C_2 $$
as characteristic curves.
You have a linear system for $x,y,z$ That splits into two subsystems. You can solve the system for $(y,z)$ using matrix exponential methods or just taking the t-derivative in one of the equations and combining with the other. Either way your solution is $$ y=C_1\cos t+C_2\sin t,\quad z=C_1\sin t-C_2\cos t $$ You also have $x=C_0e^t$ so you can find the constants from your initial data on the initial curve.
The solution of a linear system is defined everywhere and unique through any given point.