1.(Sections 8.1, 8.2) Consider the initial value problem
y' = t - y2 , y(1) = 2
(a) Perform one step of the Euler method with h=1.
(b) Perform two steps of the Euler method with h=1/2.
(c) Perform one step of the improved Euler method with h=1.
2. (Sections 8.1, 8.2, M5.3) Consider the initial value problem
y' = 2 y + t2 , y(0) = -1/4
(a) Perform 3 steps of the Euler method with h=1.
(b) Use the partial derivative of f with respect to y to make a statement about the stability of the problem. What behavior do you expect for the errors of the numerical method for increasing values of t ?
(c) Find the exact solution of the problem and the errors of the Euler values. Compare the behavior of the Euler values with the behavior of the exact solution for increasing t.
3.(Section 8.6) Consider the initial value problem
y'' + y = 1 , y(0) = 2 , y'(0) = -1
(a) Convert this problem to an equivalent first order system for two functions. (Example)
(b) Perform one step of the Euler method with h=1 for the first order system to get an approximation for y(1).
(c) Perform one step of the improved Euler method with h=1 for the first order system to get an approximation for y(1).
(d) Compare the two approximations for y(1) from (b), (c) with the exact value for y(1). Based on this information, try to predict the error of the Euler method for y(1) for h=1/2, h=1/4. Try to predict the error of the improved Euler method for y(1) for h=1/2, h=1/4.
4.(Section 8.6) Consider the initial value problem
y'' = t - yy' , y(2) = 3 , y'(2) = -4
(b) Perform one step of the Euler method with h=1/2. What is the resulting approximation for y(2.5)?
(b) Perform one step of the Improved Euler method with h=1/2. What is the resulting approximation for y(2.5)?
(c) Write a Matlab program which prints out an approximation for y(8) and plots y(t) for 2≤t≤8. Do NOT print or plot anything else!