It should be noted that the coefficients of these equations
represent the columns of the transition matrix. We find the differential
equations by defining the limit of the ratio
which yields

P_{0} ’(t) =
 l P_{0}(t) + m
P_{1}(t) 
(5.108) 
P_{1} ’(t)
= l
P_{0}(t)  m P_{1}(t)
The above equations are
called differential  difference equations.
If we say that at time t = 0 the
system was in operation, the initial conditions are P_{0}(0) = 1, P_{1}(0) = 0. It is also of interest to consider the case
where we begin when the system is down and under repair. In this case, the
initial conditions are P_{0}(0) =
0, P_{1}(0) = 1.
Transforming Equation [5.108] into
LaPlace transforms under the initial conditions
that P_{0}(0) = 1,
P_{1}(0) = 0 we have
sP_{0}(s) 
1 + l P_{0}(s)  m
P_{1}(s) =
0
sP_{1}(s) 
l P_{0}(s) + m P_{1}(s) =
0
and simplifying

(s + l)
P_{0}(s)  m P_{1}(s) =
1
(5.100) l P_{0}(s) + (s + m)
P_{1}(s) = 0 
(5.109) 
Solving these simultaneously
for P_{0}(s) yields
or
where:

s_{1} = 0 and s_{2
} = ( l + m). 

Therefore,
or, taking the inverse Laplace
transform
P_{0}(t) = L^{1}
[P_{0}(s)]