I am not getting anywhere on this problem

Let $\displaystyle u$ be a solution of

$\displaystyle

a(x,y)u_x + b(x,y)u_y = -u

$

of class $\displaystyle C^1$ in the closed unit disk $\displaystyle \Omega$ in the $\displaystyle xy$-plane. Let $\displaystyle a(x,y)x + b(x,y)y>0$ on the boundary of $\displaystyle \Omega.$ Prove that $\displaystyle u$ vanishes

identically. (Hint: Show that $\displaystyle \max_{\Omega} u \le 0, \, \min_{\Omega} u \ge 0,$ using conditions for a maximum at a boundary point.)

Since we have been using method of characteristics, I assume we need to use that.

So let $\displaystyle \Gamma: x = s,\, y = 0, \, z = w.$ Then the characteristic ODE is

$\displaystyle

\frac{dx}{dt} = a(x,y)

$

$\displaystyle

\frac{dy}{dt} = b(x,y)

$

$\displaystyle

\frac{dz}{dt} = -z

$

Therefore, $\displaystyle z = c e^{-t}$. So if $\displaystyle c > 0$ then $\displaystyle u$ is a strictly decreasing function with respect to $\displaystyle t$. If $\displaystyle c < 0$ then $\displaystyle u$ is a strictly increasing function with respect to $\displaystyle t$. However, I am not sure how this even helps me.

EDIT: I will even take some brain storming from people. I could use any help offered.