Putnam 2010 A3

Suppose that the function $h:\mathbb{R}^2\to \mathbb{R}$ has continuous partial derivatives and satisfies the equation $$h(x,y) = a \frac{\partial h}{\partial x}(x,y) + b \frac{\partial h}{\partial y}(x,y)$$ for some constants $a,b$. Prove that if there is a constant $M$ such that $|h(x,y)|\leq M$ for all $(x,y) \in \mathbb{R}^2$, then $h$ is identically zero.