I am trying to solve a problem using Method of Characteristics but I'm having trouble solving it. The question is:
Suppose $u(x,y)$ is a smooth function which is constant on any curve of the form $y=x^2+2x+C$. The value $u$ takes on different curves (parametrized by $C$) may be different. What PDE does $u$ solve?
Now I know that the Characteristics are given by these functions. The functions can be rewritten: $$C=y-x^2-2x$$Thus, given a PDE of the form: $$a(x,y)u_x+b(x,y) u_y=c(x,y)$$ We can find the Characteristic Equations: $$\dot x(s) = a(x,y), \>\>\dot y(s) = b(x,y), \>\> \dot z(s)=c(x,y)$$ Where $z(0) = u(x_0,y_0), \>\> x_0=x(0), y_0=y(0)$. But I'm quite confused by how to approach this question. If anyone could guide me in the right direction it would be appreciated. Thanks!