Solve: $xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$

Question

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?

Answer 1

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.