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Exercise 1. In Lab 2, we used the Euler and improved Euler methods to approximate y(2) for the solution of y - 2y = 3e²×, y(0) = 0 with various values of h (h = 0.1, 0.05, 0.033 0.025). Use the Runge-Kutta method with h = 0.1 to approximate y(2). How does this compare with the Euler and Improved Euler methods?
Hand In: A printout of your worksheet for the Runge-Kutta method, and your answer to the question.

Exercise 2. Consider the initial-value problem: y' = (x + y)/x, y(1) = 1.
(a) Use the Runge-Kutta method with h = .5 to approximate the solution's value at x = 2.
(b) Solve the equation by hand and compare the actual value y(2) to the approximation obtained in (a). (Your approximation should be within .001 of the actual value; if not, you must have made a mistake - find it!)
Hand In: A printout of your worksheet for the Runge-Kutta method in (a), and details of your solution by hand in (b).
Numerical methods can be misleading when singularities are involved: this is one good reason for being able to solve equations by hand. The next exercise illustrates this fact.

Exercise 3. Consider the initial-value problem: y' = y², y(0) = 3/2.
(a) Using the Runge-Kutta method with h = .5, what value do you get as an approximation for the solution at x = 1? Are you suspicious that something is wrong?
(b) Try using DIFFS to investigate the behavior of the solution. What do you suspect is going on?
(c) Find the actual solution by hand. Now can you explain what has happened? Hand In: A printout of your worksheet in (a), and an explanation (citing your experiences in (b) and (c)) for what is going on.

Exercise 4. Consider the initial-value problem y' = x² + cosy, y(0) = 0.
(a) Use Runge-Kutta with h = 0.5 and h = 0.25 to approximate the value y(1). Give y(1) to as many decimal places as you feel is accurate.
(b) Use Runge-Kutta with h = 0.1 to approximate the value y(1), and give the value to as many decimal places as you feel is accurate.
Hand In: A printout of your worksheet for the Runge-Kutta method with h = 0.5, 0.25, 0.1, and the values of y(1) requested in (a) and (b).

Exercise 5. Consider the initial-value problem y = x² + cosy, y(0) = 0 as in Exercise 4.
(a) For h = 0.5, how accurate was your value for y(1)? (I.e. E(0.5) < (what number?).)
(b) Based on your answer in (a), what accuracy would you expect for your value of y(1) when h = 0.25? (I.e. what value do you expect for E ((0.25)?) Was this accuracy achieved? (I.e. compare your values of y(1) between h = 0.25 and h = 0.1.
(c) Based on your answer in (a) and/or (b), what accuracy would you expect for your value of y(1) when h = 0.1? (I.e. without performing any calculations for h smaller than 0.1, what value do you expect for E(0.1)?)
Hand In: Give answers (in complete sentences) to the questions asked in (a), (b), and (c). Be sure to explain how you obtained your numerical values for E.

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