Chapter 7.2, Problem 1CP

Use finite differences to approximate solutions to the linear BVPs for $n=9$, $19$, and $39$.

a. $\{\begin{array}{c}{y^{\prime }}^{\prime }=y+\frac{2}{3}{e}^t\\ y\left(0\right)=0\\ y\left(1\right)=\frac{1}{3}e\end{array}$

b. $\{\begin{array}{c}{y^{\prime }}^{\prime }=\left(2+4t^2\right)y\\ y\left(0\right)=1\\ y\left(1\right)=e\end{array}$

Plot the approximate solutions together with the exact solutions (a) $y\left(t\right)=t{e}^t/3$ and (b) $y\left(t\right)=e^{t^2}$, and display the errors as a function of $t$ in a separate semilog plot.