Example 6.4 taken from S. M. Ross "Introduction to Probability Models", 10th edition.
Let $X(t)$ denote the population size at time $t$ of a birth/death process with rates:
$$ \begin{split} \mu_n &= n \mu &\quad n\geq 1\\ \lambda_n &= n \lambda+ \theta &\quad n\geq 0 \end{split} $$
Suppose that $X(0)=i$ and let $M(t) = E[X(t)]$ Consider $M(t+h)$, conditioning on $X(t)$ yields: $$ M(t+h) = E[X(t+h)] = E[E[X(t+h)|X(t)]] $$
Given the size of the population at time $t$ then, ignoring events with probability $o(h)$, the population at time $t+h$ will either increase by $1$ if a birth or an immigration occurs, decrease by $1$ if a death occurs or remain the same if neither occurs. That is: \begin{equation} X(t+h) = \begin{cases} X(t)+1, & \text{with probability } [\theta+X(t)\lambda]h+o(h) \\ X(t)-1, & \text{with probability } X(t)\mu h+o(h) \\ X(t), & \text{with probability } 1-[\theta+X(t)\lambda+X(t)\mu]h+o(h) \\ \end{cases} \tag{1} \end{equation}
Therefore: $$ E[X(t+h)|X(t)] = X(t) + [\theta+X(t)\lambda-X(t)\mu]h +o(h) \tag{2}$$
The example goes on but my question is related to these lines. The question is: starting from the information on probabilities given in $(1)$ what are the steps to end at $(2)$. I clearly see that quantities in $(2)$ comes from $1$ but I don't understand the way in which they are combined. Thanks!
The expectation is the sum of the values weighted by their probabilities. The three probabilities add to $1$ (as they must), and $X(t)$ occurs in all three values, so that yields the term $X(t)$. The remaining terms arise from the terms $\pm1$ in the values in the first two cases.