I am working on a problem where I have to prove that $P_{20}(t)=P_{10}^2(t)$, given that I have a linear Birth and death process: i.e. $\lambda_n=n.\lambda$ and $\mu_n=n.\mu$. I think the solution might be in term of using the Backward equation $P'(t)=G.P(t)$, and then integrating for the $P'_{20}(t)$ entry.
Another way might be to actually compute the $P(t)=e^{tG}$ equation and find that the relation holds between the $\{2,0\}$ and the $\{1,0\}$ entries. However I'm a bit confused on how I should find a "beautiful" convergent sum: of $e^{tG}=I+tG+t^2G^2/2+ ...$, given that my infinitesimal $G$ matrix is infinite.
Linear birth-death processes are additive, this means that the sum of some independent processes with initial populations $i$ and $j$ has the distribution of a single process with initial population $i+j$.
In particular, the probability that the latter dies before time $t$ is the probability that the former both die before time $t$. Thus, $P_{i+j,0}(t)=P_{i,0}(t)P_{j,0}(t)$, the product in the RHS following from the independence of the processes with initial populations $i$ and $j$.
Your case is when $i=j=1$.