Question:
Let $N(t)$ be a pure birth process for which
$$P(N(t+h)-N(t)=1\mid N(t)\ \text{is odd})=\alpha h+o(h)$$
$$P(N(t+h)-N(t)=1\mid N(t)\ \text{is even})=\beta h+o(h)$$
Determine $E[N(t)]$.
What I have done:
First of all, I know that this is a duplicate of this question, but there was never an actual answer given to that question, and I don't really understand all of the hint that was given in the comments.
With that being said, I do understand the first part of the hint. If we let $$A(t) = P(N(t) \;\text{even}), \;\;\;B(t) = P(N(t) \;\text{odd})$$ then I have managed to show that $$A^{'}(t) = -\beta A(t) + \alpha B(t), \;\;\;B^{'}(t) = -\alpha B(t) + \beta A(t)$$ and then I used that together with $A(t) + B(t) = 1$ to solve and found $$A(t) = \frac{\alpha}{\alpha + \beta} + A_{0}e^{-(\alpha+\beta)t}, \;\;\;B(t) = \frac{\beta}{\alpha + \beta} + B_{0}e^{-(\alpha+\beta)t}$$ Since $N(0) = 0$, we have $A(0) = 1$ and $B(0) = 0$ so $A_0 = \frac{\beta}{\alpha+\beta}$ and $A_0 = -\frac{\beta}{\alpha+\beta}$
Where I am stuck:
However, the part I don't understand is if we let $$C(t) = E[N(t)|N(t) \;\text{even}], \;\;\; D(t) = E[N(t)|N(t) \;\text{odd}]$$ then the hint claims that $$C^{'}(t) = -\beta C(t) + \alpha(C(t) + A(t)), \;\;\; D^{'}(t) = -\alpha D(t) + \beta(D(t) + B(t))$$ and I cannot intuitively see why this is true nor can I find a derivation of these equations. In addition, the user who commented this hint then says in a later comment that there is a typo in these hints. I'm hoping someone can steer me in the right direction and show me how I might derive this second set of differential equations. Once I know how to derive them I know I can solve them, I just don't know how to come up with the equations in the first place. Any help would be greatly appreciated.