Consider the IVPs

$\frac{dy}{dt}=f(t,y),atb,y(a)=$

$\frac{dy}{dt}=-12y,0t1,y(0)=1$

Write a Matlab function called RK$2$ that solves $(1)$ using Order $2$ Runge$-$Kutta method. Use this method to solve the IVP $(2)$ with $N=20,50,100.$ Call y$0,$ y$1,$ y$2,$ the three results and plot the three solutions on the same graph. Make a $loglog$ plot of absolute error at $t=1$ versus the number of intervals for all three methods on the same plot. Compare the solution at $t=1$ with the exact solution at $t=1.$ Create Err a vector that stores the absolute error of the three solutions at $t=1.$ Comment on the result $($is the method converging or not, etc.$)$ using $\%.$

Write a Matlab function called RK$4$ that solve $(1)$ using Order $4$ Runge$-$Kutta method. Use this method to solve the IVP $(2)$ with $N=20,50,100.$ Call y $3,y4,y5,$ the three results and plot the three solutions on the same graph. Make a $loglog$ plot of absolute error at $t=1$ versus the number of intervals for all three methods on the same plot. Compare the solution at $t=1$ with the exact solution at $t=1.$ Create Err$1$ a vector that stores the absolute error of the three solutions at $t=1.$ Comment on the result $($is the method converging or not, etc.$)$ using $\%.$