I would like to solve difference equations for a pure-birth process, where the rate of adding new nodes depends on the fraction of population.
$$x_i(t+1)-x_i(t)=\alpha \frac{x_i(t)}{\sum{x_j(t)}}+\beta_i \frac{x_{i-1}(t)}{\sum{x_j(t)}}$$
where $x_i$ is the population of species of kind $i$ therefore $\frac{x_i(t)}{\sum{x_j(t)}}$ represents the fraction of population at time $t$.
What I am interested in is to find a closed form solution for $x_i(t)$.
I can see a resemblance to Urn model but wasn't able to find an explicit method for solving Urn model as difference/differential equations.
I would appreciate it, if you could guide me on how to find a closed form solution for this problem or introduce references.
EDIT: $i \in [0,1,...,L]$ where $L$ is a parameter of the model. I can assume either $x_{-1}=0$ or $x_{-1}=x_{L}$ whichever simplifies the set of equations.