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Let $n$ be a positive integer and $g:\textbf{R}^n\longrightarrow \textbf{R}$ be a smooth function. Let $\partial_i$ denotes the partial derivative with respect to the the $i$th coordinate ($i\in\{1,...,n\}$). What are the functions $f:\textbf{R}^n\longrightarrow \textbf{R}$ solutions to the following partial differential equation $$ (\partial_1 \cdots \partial_n)f=gf \quad ?$$ Or, at least, what is the dimension of the $\textbf{R}$-vector space of solutions?

Many thanks!

Ali Taghavi
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Stabilo
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  • If $g(x_1, \ldots, x_n) = g_1(x_1) \cdots g_n(x_n)$ then we have the solutions $f(x_1, \ldots, x_n) = C e^{G_1(x_1)} \cdots e^{G_n(x_n)}$ where $G_i' = g_i$ and $C$ is a constant. – md2perpe Oct 05 '17 at 21:21
  • @md2perpe: Yes thank you ! But in my case, the one that led me to this question, $g$ is not separated. – Stabilo Oct 06 '17 at 17:42

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