Solve: $xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$ In the plane find two solutions of the initial-value problem
$xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$.
I think we get to use the method of characteristics But I am not much familiar how to get started? Could you please guide me?

I would suggest you use an Ansatz of the form
\begin{equation}
 u(x,y) = A x^2 + 2 B x y + C y^2 + D x + E y + F,
\end{equation}
and try to determine possible values for $A$ to $F$ using the initial condition and the PDE. The method of characteristics would give you the same result, but you'll have to use the fully nonlinear version.
