Consider the $2-$point BVP:

ODE: $$ y$+(4$x$2+2)$y $=2$x$(1+2$x$2),0<$ x $<1$; BCs: y$(0)=1,$ y$(1)=1+$ e$.$

$($a$)$ Show that y$($x$)=$ x $+$ ex$2$ is the exact solution.

$($c$)$ Use the tridiagonal solver you created in class and the given sample code on Blackboard to solve the above $2-$point BVP using the second$-$order finite difference operator as discussed in class:

$($i$)$ Let the mesh size be h $=2$p$.$ For each p $=3,4,5,6,$ plot the exact solution y$($xi$)$ and the approximate solution u$($xi$)$ for i $=1,2,,1/$h $1$ on the same plot; use the subplot in MATLAB to put all your plots on the same page;

$($ii$)$ Let the mesh size be h $=2$p$.$ For each p $=3,4,5,6,20,$ produce a table with the following data. Column $1$: h; Column $2$: $\backslash |$ uh $$ yh$\backslash |\backslash$infty ; Column $3$: $\backslash |$ uh $$ yh$\backslash |\backslash$infty $/$h$2$; Column $4$: cpu time;

Column $5$: $($cpu time$)/$N $,$ where h $=1.$ Discuss and explain the trends in each column.