AE/ME 330 Spring 2013, Homework VIII, Due Friday May 10th. 1. dy dt = te3t ?? 2y; 0 t 1; y(0) = 0 Approximate the solution to the above initial value problem using (a) Modied-Euler Method (b) Midpoint Method (c) Heun’s Method (d) 4-stage Runge-Kutta Method with a time step of h = 0:1. For each method, tabulate the approximate solution and the error at each time step using the exact solution to this problem y(t) = 15te3t ?? 1 25e3t + 1 25e??2t Plot the approximate solutions and the exact function for comparison. In another graph, plot the error distributions. Note that you may need to give a separate plot for the error of 4-stage Runge-Kutta scheme to see the trend in a larger scale. 2. dy dt = yt ?? yt 2 ; 1 t 2; y(1) = 1 Approximate the solution to the above initial value problem using (a) 2-step Adams-Bashfort Method (b) 3-step Adams-Bashfort Method with a time step of h = 0:1. In each case use starting values obtained from 4-stage Runge-Kutta method. For each method, tabulate the approximate solution and the error at each time step using the exact solution to this problem y(t) = t 1 + ln(t) Plot the approximate solutions and the exact function for comparison. In another graph, plot the error distributions. 1