Let $(X_t)$ be a continuous-time Markov process with two states, as shown below. Assume that there are two positive numbers $a$ and $b$ such that for all times $t\geq 0$ and $h>0$,
$P(X_{t+h} = 2 | X_t = 1) = ah + o(h)$
$P(X_{t+h} = 1 | X_t = 2) = bh + o(h)$
Assume that the process starts in State $1$. Let $f(t) = P(X_t = 1)$ and $g(t) = P(X_t = 2)$.
What I'm trying to do here is write out the explicit Kolmogorov forward differential equation relating $f'(t)$ and $g'(t)$ to $f(t)$ and $g(t)$ and the constants $a$ and $b$.
However, any information I find teaching the Kolmogorov forward equation is a bit difficult to understand or doesn't apply to the information I'm given above. Just wondering if anyone could help me to understand how to approach/solve the problem or any good pages on the Kolmogorov forward equation that could help lead me to answering this.