I am trying to solve the following PDE system:
I am searching for a function $f:{\mathbb{R}^+}^n\to\mathbb{R}^+$ with $n$ positive, real variables that satisfies the following PDE:
$\frac{\partial f}{\partial x_k}(x_1,...,x_n)=\frac{1}{n}\frac{f(x_1,...,x_n)}{x_k}$ for each variable $x_k$.
Even in the case of $n=2$, Mathematica's DSolve is failing me. How can this system be solved?
First consider the functions $f_j(x) = \left(x_j\right)^{\frac{1}{n}}$ for any fixed $1\leq j \leq n.$ Notice that $f_j$ is a solution to equation $j$ but not to the others. Can you think of a way of combining these?
Solution:
Try the candidate solution $$f(x) = \prod_{j=1}^n f_j(x) = \left(\prod_{j=1}^n x_j\right)^{\frac{1}{n}}.$$ Verify that $$\frac{\partial f}{\partial x_k} = \frac{1}{n} (x_k)^{\frac{1}{n} - 1} \left(\prod_{j=1, j\neq k}^n x_j\right)^{\frac{1}{n}} = \frac{1}{n} \frac{f(x)}{x_k}.$$
