In this question I tried to use the Chapman-Kolmogorov Forward equation as well as the law of total probability to solve for the Probability function
Using the fact that the generator matrix would be $$A= \begin{pmatrix} -2t & 2t \\ 2t & -2t \end{pmatrix}$$
Thus the forward differential equation would be $P^\prime_{AA}(t) = P_{AB}(t)\sigma_{BA} + P_{AA}(t)\sigma_{AA}$
which then is $P^\prime_{AA}(t) = 2t P_{AB}(t) - 2tP_{AA}(t)$
now using the fact that $P_{AA}(t)= 1-P_{AB}(t)$ then, $P_{AB}(t)= 1-P_{AA}(t)$
we can rewrite the forward differential equation as $P^\prime_{AA}(t) = 2t(1-P_{AA}(t)) - 2tP_{AA}(t)$
Solving this differential equation using the Integrating factor method gives me a solution which is not the same as the one provided. With my method I get $P_{AA}(t)=\frac {e^{-2s^2+C}}{2}+\frac{1}{2}$. Using the given fact that $P_{AA}(0)=1$, I then get $1=1$.
The solution provided is that the result is the occupancy probability $P_{AA}(t)= P_{\overline AA}(t))=e^{-s^2}$
I am not sure why my method didn't work when it should work as there are only 2 states. Any help is appreciated and sorry if this question is extremely basic and if my formatting is bad (first time posting a proper question).
ALSO If someone can help explain part B to me that would be much appreciated thank you.