Write a system of differential equation for the probabilities that the system is in state A, B, or C after time t. Consider the following process. A system is in one of the three states A, B, or C. A Poisson process is run with rate $\lambda$. At each event in this process, the following happens:
If the system is in state A, it moves to either B or C, chosen at random.
If it is system is in state B, it moves to state C.
If it is system is in state C, it moves to state A, B or C with probability 0.2, 0.3, or 0.5.
Write a system of differential equation for the probabilities that the system is in state A, B, or C after time t.
Can anyone suggest me how we think in this question. I never solve this type of question before.
 A: The first thing to do is deal with the rate $\lambda$. This is just a constant that tells what is the number of transitions per unit time. I prefer to think of it as the constant that allows me to write dimensionless equations. So if $\frac {dP}{dt}$ has units of $s^{-1}$, and $\lambda$ has the same units, then $\frac {dP}{\lambda dt}$ is dimensionless.
If the only transitions are described by the first equation, the change in probability for the $A$ state would be $$\frac{dP_A}{\lambda dt}=-P_A$$
This means that everything that was $A$ should be gone. These would transform into $B$ or $C$ state, with equal probability. This can be written as:$$\frac{dP_B}{\lambda dt}=0.5 P_A\\\frac{dP_C}{\lambda dt}=0.5 P_A$$
Now consider the effect of the transition from $B$ to $C$. That transition says that $P_B$ will decrease, and the rate is $-P_B$. Meanwhile $P_C$ will increase at the rate $P_B$. So now the three equations can be written as
$$\begin{align}\frac{dP_A}{\lambda dt}&=-P_A\\ \frac{dP_B}{\lambda dt}&=0.5 P_A-P_B\\\frac{dP_C}{\lambda dt}&=0.5 P_A+P_B\end{align}$$
The last is the transition from $C$. The rate of transition to $A$ will be $0.2P_C$, to $B$ will be $0.3P_C$. But you can also transition from $C$ to $C$. The rate for this will be $0.5P_C$. Note however that you need to decrease $P_C$ by all the transitions to the same or other states. We can therefore write the final equations as:
$$\begin{align}\frac{dP_A}{\lambda dt}&=-P_A+0.2P_C\\ \frac{dP_B}{\lambda dt}&=0.5 P_A-P_B+0.3P_C\\\frac{dP_C}{\lambda dt}&=0.5 P_A+P_B+(0.5-1)P_C\end{align}$$
