Implement a fourth$-$order Runge$-$Kutta method that integrates the system $y^{}=f(y)$ from $t=0$

to $t=100$ with stepsize $h=0\mathrm{.}05.$ Use the four initial conditions described in the table below.

You can use the pendplot function to animate the motion of the pendulum, which might be

fun and might help you debug your code. But you do not need to include these plots in your

report.

For each case, plot the function ${}_2(t)$ versus time. Cases $3$ and $4$ demonstrate the initial value

sensitivity of the system, namely, that a small perturbation can lead to drastically different

solutions.

Your report should contain the MATLAB commands required to produce the results, and the

four plots of ${}_2(t).$

Run Case $1$ in problem $,$ with the five stepsizes $h=\frac{0\mathrm{.}05}{2^{(k-1)}},k=1,2,3,4,$ and $h=0\mathrm{.}001.$

Compute the value of ${}_2(t=100)$ for each stepsize. Consider the last result the exact solution,

and plot the four errors as a function of $h$ in a loglog$-$plot. Estimate the order of convergence

from the slope.

Your report should contain the MATLAB commands used, the five values of ${}_2(t=100),$ the

loglog$-$plot, and the estimated slope.