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.