Consider a birth and death process with the birth rate $\lambda_m = \lambda (m\ge 0)$ and death rate $\mu_m = m \mu (m \ge 1)$.
A. How would I derive the stationary distribution?
Only information I have is this from my notes: 
B. Assuming $X(t)$ is the state at time $t$, how would I derive the equation for $M(t) = E[X(t)]$ using differential equations?
I know that for this problem I need to use queuing system $M/M/\infty$ and use the identity $M(t+h) - M(t) = E[E[X(t+h) - X(t) \mid X(t)]]$ but where do I go from here?
C. Solve the equation for $E[M(t)]$ and show the limiting value of this expectation at $t\to\infty$ equals to $\lambda / \mu$
Is there a formula I can use to figure out these problems?
The detailed balance equations give $\mu (m+1)\pi_{m+1}=\lambda \pi_m$. this has solution $\pi_m=\frac{\rho^m}{m!}\pi_0$. (where $\rho=\lambda / \mu$) Total probability is one so sum to find $\pi_0$=$e^{-\rho}$. This is the stationary solution.
The next part is a bit harder. Write down the forward equations which gives you a differential difference equation in $p_m(t)$ Define the generating function $G(z,t)=\sum p_m z^m$. Combine the two to get a partial differential equation for G. Differentiate this wrt z and note $M(t)=\frac{dG}{dz}$ evaluated at z=1. This gives you an easy equation to solve for M(t) which has the desired limit and agrees with stationary solution in part 1.
Can't help feeling there is an easier way!
And there is. Multiply the forward equation through by m and sum over m. After rejigging the summation this gives a simple differential equation in M.