Modified-euler method

Question

1. dy/dt = te^3t - 2y, 0 ≤ t ≤ 1, y(0) = 0

Approximate the solution to the above initial value problem using

(a) Modified-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) = 1/5 te^3t -1/25 e^3t +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 = y/t- (y/t)^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.