Let $u(x,y)$ be a continuously differentiable function on the closed unit disc and is a solution to $a(x,y)u_x+b(x,y)u_y=-u,$ on the closed unit disc. Suppose $a(x,y)x+b(x,y)y>0,$ on the boundary of the closed unit disc. $a,b$ are given smooth functions.
Prove that $u$ vanishes identically zero.