# What are the solutions of the equation $(\partial_1 \cdots \partial_n)f=gf$?

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!

• 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 5 '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 6 '17 at 17:42