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:
(1) p.7-21
According to p.7-20, the general form of the series is:
![{\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)
(2) p. 7-20
From (2) p.7-20, we can obtain n+1 equations for n+1 unknown coefficients 
.
After referring to p.7-22, it can be determined that the matrix to be set up is of the following form:
where the rows signify the coefficients 
,
and the columns signify 
.
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 
:
j=0: 
(1)
Equation associated with 
:
j=1: 
(2)
Equation associated with 
:
j=2: 
(3)
Equation associated with 
:
j=n-2: ![{\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)
(n-2)
Equation associated with 
:
j=n-1: 
(n-1)
Equation associated with 
:
j=n: 
(n)
Using all of the above equations, (1), (2), (3), (n-2), (n-1), (n), we can then determine the A matrix to be:
