What is the difference between the forward and backward equations in a CTMC?

Question

Given that the Forward equation in a CTMC (Continuous Time Markov Chain) is: $P'(t)=P_t G$, and the Backward equation is: $P'(t)=G P_t$, which equations should I use of the two depending on the case I am studying? All I see in the literature is "this is the forward equation, and this is the backward equation", with no practical examples on where I should use them, or use one over the other.

Answer 1

You solve both of them with: $$P(t)=\exp(tG)$$

Answer 2

Both equations give you the dynamics of the system. The only difference is in the order in which the elementary transitions happen. I think that you may be free to choose the most convenient case. Usually The forward equation is more used.

The stationary solution of both equations is the same under certain broad conditions (for example it is possible show that for a finite state Markov process this property is always satisfied).