Consider the following equation in two variables $$a_1x_1x_2+a_2x_2=c$$
where $x_i$ are variables and the other constants can be any real numbers. In three variables, this is $$a_1x_1x_2x_3+a_2x_2x_3+a_3x_3=c$$
In $n$ variables the equation becomes
Solve the equation in $n$ variables for the $x_i$.
The equation doesn't seem to complicate, but the only way to solve it I see, is to use numerical methods. The gradient and Hessian are not difficult to compute, so Newton-Raphson can be used.
Is there a way to analytically find the solution of the equation?
If numerical method are the only way to go, as the equation doesn't seem too complicate, is it possible to guaranted all roots have been found?
In my case $n=30$ and the $x_i$ are in $(0,2)$.