346
Chapter
9
S - U - ^ Z -
s + KsV
Pumx
(
6
)
one obtains
X
i
*p
a
fp +
^max
^0
■ ( j r - i - j r 0)JT
(
7
)
where
X
max
P
m o .
P p
V o
+ x n
(
8
)
The standard technique for integration of a differential equation of the type of Eq. (3) is
1.
Use synthetic division to separate the original expression into a number of terms of zero or higher
degree in
z
(or
X)
and a final term of the form of Eq. (3) in which the numerator polynomial
P
’(z)
has a degree
np
<
n
q.
2.
The first terms can be integrated directly. The last term is decomposed into a number of partial
fractions, one for each zero in
Q(z)
[or
Q{X)\
When a zero z, is of degree
n,
it gives rise to
n
fractions of the form
(z -
z()"", (z - z, ) 'n+l,.
.., (z - z,)~‘. The final result is
P(z)
-------- =
Ü
xQ{P)
z
+ ÜUZ
+
a.
A
+ — +
z
z - z
A,
( z - z X
+ *” (9)
3.
The constants Ao , A |, .
.., A,i , Am
,... are determined by identification of the coefficients to equal
powers of z in
P \z)
and on the right-hand side of Eq. (9)
4.
Each of the resulting terms is easily integrated.
As examples of the procedure, consider first Eq. (5) and next Eq. (7):
£(l + ^o
- X ) 2 +(1+X0 -X )+ a
_
| ^
1
^ i 1
+ X Q
- X + a
X(\ + X Q- X )
~
1
o) X
X(1 + X Q-X )
6(1 +
X
o )------
b
+ ——
+
A
1 +
X n - X
(10)
0 X
X(l + X 0 - X )
or
Ai
-
A0
= -1 and
A
(1 +
X ü)
= 1 + X0 +
a
. Hence the solution of Eq. (5) is