STAT 333
Solutions To Differential Equations
Suppose
x
=
x
(
t
) is a function of
t
. Then:
1.
x
prime
=
a x
has general solution
x
=
c e
at
where
c
is any constant.
2.
x
prime
=
a x
+
f
(
t
) has general solution
x
=
e
at
{
c
+
integraltext
t
0
e

as
f
(
s
)
ds
}
for any constant
c
.
Application to Solving for P
(t) In The TwoState Chain
In a twostate Markov chain the generator matrix is given by
Q
=
parenleftbigg

λ
λ
μ

μ
parenrightbigg
where
λ
and
μ
can be any positive constants. The Kolmogorov Forward Equations are given
by:
P
prime
(t) = P
(t) Q
.
We will solve these equations. Now since
P
00
(
t
) +
P
01
(
t
) = 1 and
P
10
(
t
) +
P
11
(
t
) = 1, there
are only two unknowns
P
00
(
t
) and
P
11
(
t
).
In pointwise form the Forward Equation for
P
prime
00
(
t
) looks like
P
prime
00
(
t
) =

λP
00
(
t
) +
μP
01
(
t
)
=

λP
00
(
t
) +
μ
(1

P
00
(
t
))
=

(
λ
+
μ
)
P
00
(
t
) +
μ
This is of the form 2. where
x
=
P
00
(
t
),
a
=

(
λ
+
μ
), and
f
(
t
) =
μ
. Thus the solution is
given by
P
00
(
t
) =
e

(
λ
+
μ
)
t
{
c
+
integraldisplay
t
0
e
(
λ
+
μ
)
s
μ ds
}
=
e

(
λ
+
μ
)
t
{
c
+
μ
λ
+
μ
e
(
λ
+
μ
)
t

μ
λ
+
μ
}
=
μ
λ
+
μ
+
λ
λ
+
μ
e

(
λ
+
μ
)
t
where this last follows because
P
00
(0) = 1 =
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 Spring '08
 Chisholm
 Derivative, 1 j, twostate chain

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