I am currently working on a question and require some help to proceed. The question is:
Use Method of Characteristics for: $$y\cdot u_x-2xy\cdot u_y=2xu$$ With $u=y^3$ when $x=0$ and $1\leq y\leq 2$
Now I am trying to solve it, I have the characteristic equations: $$\dot x(s)=y(s),\>\> \dot y(s) = 2x(s)y(s), \>\> \dot z(s)=2x(s)z(s)$$ I'm not sure if these are the right general auxiliary conditions, but I put: $$x(0)=0, \>\>y(0)=y_0, \>\> z(0)=y_0^3$$ Now to solve for $x(s)$ and $y(s)$, the question says to multiply the first by $2x(s)$ and then add. I'm not sure how that would help, and am looking for some guidance. Thanks!