Euler's Method The Euler Method is the simplest of the numerical methods we cover, and as we saw in class is closely related to a Left Riemann Sum. The algorithm (in pseudo code) for Euler's Method is as follows: define f(ty) input initial values to and yo input step size h and number of stepsn output to and yo for, from 1 ton k1=f(t,y) y = y + h*k1 t = tuh end output t and y. 1. Write a script that applies Euler's Method to the initial value problem y'=1-t + 4y, with y(O)=1 on the interval[0, 2] using a step size of h=0.01. Compare your answer to the values 2. Write a script that applies the Improved Euler Method to the same function as in problem 1. Finally, we have another method that uses a weighted average of slopes at four different points to estimate the integrals, which is precisely equal to Simpson's rule iff does not depend on y. Again, only the for-loop gets changed in the algorithm. K1 = f(t, y) K2 = f(t +0.5*h, y +0.5*h*k1) K3 = f(t +0.5*h, y +0.5*h*k2) K4 = f(t + h, y + h*k3) Y = y + (h/6)*(k1 + 2*k2 + 2*k3+k4) T=t+h 3. Finally apply this method to the same problem in 1 and 2. This time use a step size of h = 0.1 and compare your results to those given in the table on p. 470. 4. Solve the initial value problem (1.5)=0.5 dty tell a. Use ode45() to find the approximate values of the solution at t = 0.1.1.8.2.1 and also plot the solution. b. Now plot the numerical solution of several large intervals and make a guess about the nature of the solution as t 00. 5. Consider the initial value problem e" + (te sin y) y = D. H(2)=1.5 a. Use ode45() to find approximate values of the solution at x =1.1.5. and 3. Then plot the solution on the interval [0.5.4]. b. Plot the solution for several large intervals and guess at the behavior of the solution as C. Plot the solution for several intervals of the form [e. 2], where is a small number. What do you think is happening as t 0*