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A3

Part of 2010 Putnam

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Putnam 2010 A3

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12/6/2010
Suppose that the function h:R2→Rh:\mathbb{R}^2\to\mathbb{R} has continuous partial derivatives and satisfies the equation h(x,y)=a∂h∂x(x,y)+b∂h∂y(x,y)h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y) for some constants a,b.a,b. Prove that if there is a constant MM such that ∣h(x,y)∣≤M|h(x,y)|\le M for all (x,y)(x,y) in R2,\mathbb{R}^2, then hh is identically zero.
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