User:Egm4313.s12.team8.kyle/R4.1

The general form of (2) p.7b-13 is shown on p.7c-20 and is as follows:
 * $$\sum_{j=0}^{n-2}[c_{j+2} (j+2) (j+1) + a c_{j+1} (j+1) + bc_j]x^j

+ac_nnx^{n-1} + b[c_{n-1}x^{n-1} + c_nx^n] = \sum_{j=0}^{n} d_jx^j $$

When j=0:
 * $$2c_{2}+ac_{1}+bc_{0}=d_{0}$$

When j=1:
 * $$6c_{3}+2ac_{2}+bc_{1}=d_{1}$$

When j=2:
 * $$12c_{4}+3ac_{3}+bc_{2}=d_{2}$$

When j=(n-2):
 * $$[c_{n}(n)(n-1)+ac_{n-1}(n-1)+bc_{n-2}]=d_{n-2}$$

When j=(n-1):
 * $$ac_{n}n+bc_{n-1}=d_{n-1}$$

When j=n:
 * $$bc_{n}=d_{n}$$

Where $$c=[c_0, c_1, ..., c_{n-1}, c_n]^{T}$$ and $$d=[d_0, d_1, ..., d_{n-1}, d_n]^{T}$$

and $$Ac=d$$

The general form of matrix A can be created from the coefficients shown. $$A= \begin{bmatrix} b &     a &  2     &      0 &       0 &       0 &       0 \\ 0 &     b & 2a     &      6 &       0 &       0 &       0 \\ 0 &     0 &  b     &     3a &      12 &       0 &       0 \\ 0 &     0 &      0 & \ddots & \ddots  &  \ddots &       0 \\ 0 &     0 &      0 &      0 &       b & a (n-1) & n (n-1) \\ 0 &     0 &      0 &      0 &       0 &       b & a n     \\ 0 &     0 &      0 &      0 &       0 &       0 &   b \end{bmatrix} $$