What is the difference between the forward and backward equations in a CTMC? 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.
 A: Suppose you have a particle which you know its position and velocity at $t=0$ and the force affecting it at $t > 0$. So you can evolve the position and velocity of the particle to any point in future (usually by a differential equation).
This is the analogous to a Forward Equation, but on deterministic setting (Kolmogorov himself used this example in his article On Analytical Methods in the Theory of Probability).
Sometimes you actually have the final condition and want to work out its inicial. It is common in financial derivatives, where you know your payoff function at maturity, have a model for the dynamics of its underline asset and want to know the inicial price of the derivative. 
So you need to work in reverse, this is the Backward Equation.
Bottom line: if you have a markov probabilistic setting, where you know the inicial condition and its dynamics, work out the Forward Equation; if the know is the final condition, work out the Backward Equation. 
A: You solve both of them with: $$P(t)=\exp(tG)$$
A: 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).
