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 $$ E[X_t]=\frac{\partial u}{\partial s}(t,1). $$