Chapter 7.2, Problem 2CP

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

a. $\{\begin{array}{c}9{y^{\prime }}^{\prime }+{\pi }^{2}y=0\\ y\left(0\right)=-1\\ y\left(\frac{3}{2}\right)=3\end{array}$

b. $\{\begin{array}{c}{y^{\prime }}^{\prime }=3y-2y^{\prime }\\ y\left(0\right)=e^3\\ y\left(1\right)=1\end{array}$

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