a) Find the general solution of the equation
$\displaystyle (x^2 - y^2 - u^2)u_x + 2xyu_y = 2xu$
b) Find the particular solution when
$\displaystyle u(x,1)=x$.
The characteristic system is $\displaystyle \begin{cases} dx/(x^2-y^2-u^2)=dy/2xy \ (1)\cr du/dy=u/y \ \ \ (2)\end{cases}$.
Equation (2) gives $\displaystyle u=cy$, so substituting into (1) we obtain
$\displaystyle dx/dy=(x/2y)-[(1+c^2)y/2x]$. Solve this (how?) to get
$\displaystyle x^2=2y^2(2c^*y-d)$, where $\displaystyle c^*=c^*(c)=-(1+c^2)/2$. It all eventually leads to $\displaystyle x^2/2=-(1/2)-(u^2y/2y^2x)+d$,
and use the initial conditions $\displaystyle \{y=1,u=x\}$ to determine $\displaystyle d=d(x)$.
Oh and check the calculations...