Question 11.1.2: Apply the Linear Shooting technique with N = 10 to the bound...
Apply the Linear Shooting technique with N = 10 to the boundary-value problem
y^{\prime \prime}=-\frac{2}{x} y^{\prime}+\frac{2}{x^{2}} y+\frac{\sin (\ln x)}{x^{2}}, \quad \text { for } 1 \leq x \leq 2, \text { with } y(1)=1 \text { and } y(2)=2,
and compare the results to those of the exact solution
y=c_{1} x+\frac{c_{2}}{x^{2}}-\frac{3}{10} \sin (\ln x)-\frac{1}{10} \cos (\ln x) ,
where
c_{2}=\frac{1}{70}[8-12 \sin (\ln 2)-4 \cos (\ln 2)] \approx-0.03920701320
and
c_{1}=\frac{11}{10}-c_{2} \approx 1.1392070132
Applying Algorithm 11.1 to this problem requires approximating the solutions to the initial-value problems
y_{1}^{\prime \prime}=-\frac{2}{x} y_{1}^{\prime}+\frac{2}{x^{2}} y_{1}+\frac{\sin (\ln x)}{x^{2}}, \quad \text { for } 1 \leq x \leq 2 \text {, with } y_{1}(1)=1 \text { and } y_{1}^{\prime}(1)=0
and
y_{2}^{\prime \prime}=-\frac{2}{x} y_{2}^{\prime}+\frac{2}{x^{2}} y_{2}, \quad \text { for } 1 \leq x \leq 2, \text { with } y_{2}(1)=0 \text { and } y_{2}^{\prime}(1)=1 .
The results of the calculations, using Algorithm 11.1 with N = 10 and h = 0.1, are given in Table 11.1. The value listed as u_{1, i} \text { approximates } y_{1}\left(x_{i}\right) \text {, the value } v_{1, i} approximates y_{2}\left(x_{i}\right), \text { and } w_{i} approximates
y\left(x_{i}\right)=y_{1}\left(x_{i}\right)+\frac{2-y_{1}(2)}{y_{2}(2)} y_{2}\left(x_{i}\right) .
Table 11.1
\begin{array}{llllcc}\hline x_{i} & u_{1, i} \approx y_{1}\left(x_{i}\right) & v_{1, i} \approx y_{2}\left(x_{i}\right) & w_{i} \approx y\left(x_{i}\right) & y\left(x_{i}\right) & \left|y\left(x_{i}\right)-w_{i}\right| \\\hline 1.0 & 1.00000000 & 0.00000000 & 1.00000000 & 1.00000000 & \\1.1 & 1.00896058 & 0.09117986 & 1.09262917 & 1.09262930 & 1.43 \times 10^{-7} \\1.2 & 1.03245472 & 0.16851175 & 1.18708471 & 1.18708484 & 1.34 \times 10^{-7} \\1.3 & 1.06674375 & 0.23608704 & 1.28338227 & 1.28338236 & 9.78 \times 10^{-8} \\1.4 & 1.10928795 & 0.29659067 & 1.38144589 & 1.38144595 & 6.02 \times 10^{-8} \\1.5 & 1.15830000 & 0.35184379 & 1.48115939 & 1.48115942 & 3.06 \times 10^{-8} \\1.6 & 1.21248372 & 0.40311695 & 1.58239245 & 1.58239246 & 1.08 \times 10^{-8} \\1.7 & 1.27087454 & 0.45131840 & 1.68501396 & 1.68501396 & 5.43 \times 10^{-10} \\1.8 & 1.33273851 & 0.49711137 & 1.78889854 & 1.78889853 & 5.05 \times 10^{-9} \\1.9 & 1.39750618 & 0.54098928 & 1.89392951 & 1.89392951 & 4.41 \times 10^{-9} \\2.0 & 1.46472815 & 0.58332538 & 2.00000000 & 2.00000000 & \\\hline\end{array}