Suppose we have $2$ independently working machines which break down with intensity $\lambda > 0$ and are repaired by one mechanic with intensity $\mu > 0$. Find system od retrospective Kolmogorov equations. What is the probability that at time $t$ both machines are working if both were working at time $0$?
I know that we have \begin{cases} P_{00}'(t) = 2\lambda P_{10}(t) -2\lambda P_{00}(t) \newline P_{10}'(t) = \mu P_{00}(t) + \lambda P_{20}(t) -(\mu + \lambda)P_{10}(t) \newline P_{20}'(t) = 2\mu P_{10}(t) -2\mu P_{20}(t) \end{cases}
and that we can obtain $$ \mu^2 P_{00}(t) + 2\mu\lambda P_{10}(t) + \lambda^2 P_{20}(t) = \mu^2 $$
by cleverly multiplying and adding the three equations and then integrating and using $P_{00}(0) = 1$, but don't see how to calculate $P_{00}(t)$.