Obtain equations (2), (3), (n-2), (n-1), (n), and set up the matrix A as in (1) p.7-21
for the general case, with the matrix coefficients for rows 1, 2, 3, (n-2), (n-1), n, filled in,
as obtained from equations (1), (2), (3), (n-2), (n-1), (n).
As shown in p.7-21, the first equation is:
2
C
2
+
a
c
1
+
b
c
0
=
d
0
{\displaystyle 2C_{2}+ac_{1}+bc_{0}=d_{0}\!}
According to p.7-20, the general form of the series is:
∑
j
=
0
n
−
2
[
c
j
+
2
(
j
+
2
)
(
j
+
1
)
+
a
c
j
+
1
+
b
c
j
]
x
j
+
a
c
n
n
x
n
−
1
+
b
[
c
n
−
1
x
n
−
1
+
c
n
x
n
]
=
∑
j
=
0
n
d
j
x
j
{\displaystyle \sum _{j=0}^{n-2}[c_{j+2}(j+2)(j+1)+ac_{j+1}+bc_{j}]x^{j}+ac_{n}nx^{n-1}+b[c_{n-1}x^{n-1}+c_{n}x^{n}]=\sum _{j=0}^{n}d_{j}x^{j}\!}
From (2) p.7-20, we can obtain n+1 equations for n+1 unknown coefficients
c
0
,
.
.
.
,
c
n
{\displaystyle {c_{0},...,c_{n}}\!}
After referring to p.7-22, it can be determined that the matrix to be set up is of the following form:
A
=
[
X
X
X
0
0
0
0
X
X
0
0
0
0
0
X
0
0
0
0
0
0
X
X
X
0
0
0
0
X
X
0
0
0
0
0
X
]
{\displaystyle A={\begin{bmatrix}X&&X&&X&&0&&0&0\\0&&X&&X&&0&&0&0\\0&&0&&X&&0&&0&0\\0&&0&&0&&X&&X&X\\0&&0&&0&&0&&X&X\\0&&0&&0&&0&&0&X\end{bmatrix}}\!}
where the rows signify the coefficients
c
0
,
c
1
,
c
2
,
c
n
−
2
,
c
n
−
1
,
c
n
{\displaystyle c_{0},c_{1},c_{2},c_{n-2},c_{n-1},c_{n}\!}
d
0
,
d
1
,
d
2
,
d
n
−
2
,
d
n
−
1
,
d
n
{\displaystyle d_{0},d_{1},d_{2},d_{n-2},d_{n-1},d_{n}\!}
Building the coefficient matrix as shown in p.7-22 of the class notes, we can begin to solve for the coefficients
as follows:
Equation associated with
d
0
{\displaystyle d_{0}\!}
j=0:
d
0
=
2
C
2
+
a
c
1
+
b
c
0
{\displaystyle d_{0}=2C_{2}+ac_{1}+bc_{0}\!}
Equation associated with
d
1
{\displaystyle d_{1}\!}
j=1:
d
1
=
6
c
3
+
2
a
c
2
+
b
c
1
{\displaystyle d_{1}=6c_{3}+2ac_{2}+bc_{1}\!}
Equation associated with
d
2
{\displaystyle d_{2}\!}
j=2:
d
2
=
12
c
4
+
3
a
c
3
+
b
c
2
{\displaystyle d_{2}=12c_{4}+3ac_{3}+bc_{2}\!}
Equation associated with
d
n
−
2
{\displaystyle d_{n-2}\!}
j=n-2:
d
n
−
2
=
[
c
n
(
n
)
(
n
−
1
)
+
a
c
n
−
1
(
n
−
1
)
+
b
c
n
−
2
]
{\displaystyle d_{n-2}=[c_{n}(n)(n-1)+ac_{n-1}(n-1)+bc_{n-2}]\!}
Equation associated with
d
n
−
1
{\displaystyle d_{n-1}\!}
j=n-1:
d
n
−
1
=
a
c
n
n
+
b
c
n
−
1
{\displaystyle d_{n-1}=ac_{n}n+bc_{n-1}\!}
Equation associated with
d
n
{\displaystyle d_{n}\!}
j=n:
d
n
=
b
c
n
{\displaystyle d_{n}=bc_{n}\!}
Using all of the above equations, (1), (2), (3), (n-2), (n-1), (n), we can then determine the A matrix to be:
A
=
[
b
a
2
0
0
0
0
b
2
a
0
0
0
0
0
b
0
0
0
0
0
0
b
a
(
n
−
1
)
n
(
n
−
1
)
0
0
0
0
b
a
n
0
0
0
0
0
b
]
{\displaystyle A={\begin{bmatrix}b&&a&&2&&0&&0&0\\0&&b&&2a&&0&&0&0\\0&&0&&b&&0&&0&0\\0&&0&&0&&b&&a(n-1)&n(n-1)\\0&&0&&0&&0&&b&an\\0&&0&&0&&0&&0&b\end{bmatrix}}\!}
![{\displaystyle \sum _{j=0}^{n-2}[c_{j+2}(j+2)(j+1)+ac_{j+1}+bc_{j}]x^{j}+ac_{n}nx^{n-1}+b[c_{n-1}x^{n-1}+c_{n}x^{n}]=\sum _{j=0}^{n}d_{j}x^{j}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5b39d8b5978e7e6059a343f46c4ebdf2cf61c6)
![{\displaystyle d_{n-2}=[c_{n}(n)(n-1)+ac_{n-1}(n-1)+bc_{n-2}]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a77c2db157f275bc2f747f87c340d4c18dfc5073)
