If $$h(x,y)=a\frac{\partial h(x,y)}{\partial x}+b\frac{\partial h(x,y)}{\partial y}$$ for some $a,b \in \mathbb{R}$ and $|h(x,y)|\le M$ for every $(x,y)\in \mathbb{R}^2$ then show $h$ is identically equal to zero. I just want a hint.... I'm stuck.
Bounded function equal to sum of partial derivatives
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analysis
multivariable-calculus
1 Answers
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Fix $(x,y)\in \Bbb R^2$ and set $\phi(t) = h(x + at, y + bt)$, for all $t\in \Bbb R$. Show that $\phi$ satisfies the differential equation $\phi'(t) = \phi(t)$. Using the boundedness condition $\lvert h \rvert \le M$, deduce that $\phi = 0$, and thus $h(x,y) = 0$.