For a set of under-determined linear equations, I was wondering if there is any closed form for all non-negative solutions? Is there a way to analytically characterize the feasibility set of such equations?


More formally, for given $M\in \mathbb{R}^{n\times m}\_{+}$ and $y\in \mathbb{R}^{m\times 1}$, find all solution $x\geq 0$ of $Mx=y$. Note that $M$ and $y$ are non-negative as well. 



nonnegative solution of nonhomogeneous under-determined linear system of equations
All such vectors $x\geq 0$ form a polyhedron in $\mathbb{R}^n$, which may be empty. The latter can be certified using Farkas Lemma, as David points out in the comment. On the other hand, a nonempty polyhedron is a Minkowski sum of a subspace, a polyhedral cone, and a polytope. 
(see Wikipedia for terms I used).


Your extra conditions on $M$ and $y$ imply that you only get a polytope $P$. Then you can say that any $x$ is a convex combination of the vertices of $P$ (for practical purposes, this is often a useless kind of description, but you cannot hope for anything better in general).


