Chapter 7.2, Problem 3CP

Use finite differences to approximate solutions to the nonlinear boundary value problems for $n=9$, $19$, and $39$.

a. $\{\begin{array}{c}{y^{\prime }}^{\prime }=18y^2\\ y\left(1\right)=\frac{1}{3}\\ y\left(2\right)=\frac{1}{12}\end{array}$

b. $\{\begin{array}{c}{y^{\prime }}^{\prime }=2e^{-2y}\left(1-t^2\right)\\ y\left(0\right)=0\\ y\left(1\right)=\ln 2\end{array}$

Plot the approximate solutions together with the exact solutions (a)

$y\left(t\right)=1/\left(3t^2\right)$ and (b)

$y\left(t\right)=\ln \left(t^2+1\right)$, and display the errors as a function of $t$ in a separate semilog plot.