Mean number of particle present in the system: birth-death process, $E(X_t|X_0=i)$, $b_i=\frac{b}{i+1}$, $d_i=d$ Let $\{X_t\}$ be a birth–and–death process with birth rate
$$
b_i = \frac{b}{i+1},
$$
when $i$ particle are in the system, and a constant death rate
$$
d_i=d.
$$
Find the expected number of particle in the system at time $t$, given that $X_0=i$.
Define
$$
f(t)=E(X_t),
$$
and 
$$
p_n=P\left(X_t=n | X_0=i \right).
$$
Using the foward equation,
$$
f'(t)=\sum_{n=1}^\infty n \left( p_{n-1} \frac{b}{n} + p_{n+1}d + p_n\left( 1 - \frac{b}{n+1}-d\right)\right).
$$
After simplification, I have
$$
f'(t)=p_0 b - \sum_{n=1}^\infty\frac{b}{n+1}p_n + d + f(t),
$$
and I don't see how to solve this differential equation.

Not sure one can get explicit formulas for $E[X_t]$ but anyway, your function $f$ is not rich enough to capture the dynamics of the process. 
The canonical way to go is to consider $u(t,s)=E[s^{X_t}]$ for every $t\geqslant0$ and, say, every $s$ in $(0,1)$. Then, pending some errors in computations done too quickly, the function $u$ solves an integro-differential equation similar to
$$
\frac{s}{1-s}\cdot\frac{\partial u}{\partial t}(t,s)=d\cdot(u(t,s)-u(t,0))-b\int_0^su(t,r)\mathrm dr,
$$
with initial condition $u(0,s)=s^i$. Assuming one can solve this (which does not seem obvious at first sight), your answer is 
$$
E[X_t]=\frac{\partial u}{\partial s}(t,1).
$$
