Question 11.3.1: Use Algorithm 11.3 with N = 9 to approximate the solution to...
Use Algorithm 11.3 with N = 9 to approximate the solution to the linear 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 obtained using the Shooting method in Example 2 of Section 11.1.
For this example, we will use N = 9, so h = 0.1, and we have the same spacing as in Example 2 of Section 11.1. The complete results are listed in Table 11.3.
Table 11.3
\begin{array}{cccc}\hline x_{i} & w_{i} & y\left(x_{i}\right) & \left|w_{i}-y\left(x_{i}\right)\right| \\\hline 1.0 & 1.00000000 & 1.00000000 & \\1.1 & 1.09260052 & 1.09262930 & 2.88 \times 10^{-5} \\1.2 & 1.18704313 & 1.18708484 & 4.17 \times 10^{-5} \\1.3 & 1.28333687 & 1.28338236 & 4.55 \times 10^{-5} \\1.4 & 1.38140205 & 1.38144595 & 4.39 \times 10^{-5} \\1.5 & 1.48112026 & 1.48115942 & 3.92 \times 10^{-5} \\1.6 & 1.58235990 & 1.58239246 & 3.26 \times 10^{-5} \\1.7 & 1.68498902 & 1.68501396 & 2.49 \times 10^{-5} \\1.8 & 1.78888175 & 1.78889853 & 1.68 \times 10^{-5} \\1.9 & 1.89392110 & 1.89392951 & 8.41 \times 10^{-6} \\2.0 & 2.00000000 & 2.00000000 & \\\hline\end{array}
These results are considerably less accurate than those obtained in Example 2 of Section 11.1. This is because the method used in that example involved a Runge-Kutta technique with local truncation error of order O\left(h^{4}\right) , whereas the difference method used here has local truncation error of order O\left(h^{2}\right) .