Problem
A dolphin swims between 3 different pools, A B and C. The time is spend in each pool, before going to the next one, is Exp(1/2). The possible ways for it to travel is A to B. B to C. C to A. C to B. When in C, it choses B or A randomly with the same probability.
Given the dolphin starts in pool X at time 0 define $P_{XY}(t)$ as "the probabilitythat the dolphin is in pool Y at time t".
Find $P_{AA} (t)$ $P_{AB} (t)$ $P_{AC} (t)$.
Attempt
Since we have a finite space {A,B,C } I use chapman kolmogorov's forward equations, and obtain
$$P'_{AA} (t) = 1 \cdot (1-e^{-2t})-2 \cdot (e^{-2t})$$
$$P'_{AB} (t) = 2 \cdot e^{-2t} + 1 \cdot 2t^2e^{-2t} -2 \cdot 2te^{-2t}$$
$$P'_{AC} (t) = 2 \cdot 2te^{-2t} - 2 \cdot 2t^2e^{-2t}$$
where the third line is not really needed. And of course, $P_{AA} (t)+P_{AB} (t)+P_{AC} (t)=1$.
Question
Now we have everything needed to solve the equations, correct?