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Math. 423  Final exam  Due by Wednesday, May 10

1. Given the initial value problem y′ = \frac{cos(t) − 2yt}{1 + t^{2}} with y(0) = 0.

Use MAPLE or any other program to answer the following questions. Write the answers below and attach some documentation, such as a printout.

(a) Use the modified Euler method, first with step size h = 0.05 and then with h = 0.025, to approximate y(1).

(b) Do the same using the standard Runge-Kutta method of order four.

(Continuing problem 1) The actual solution of the initial value problem is

y(t) = \frac{sin(t)}{1 + t^{2}} .

(c) Determine the errors involved in using the modified Euler method with h = 0.05 and h = 0.025. By what factor does the error improve when the step size is reduced? Is that about what one would expect? Explain.

(d) Answer the same questions for the standard Runge-Kutta method of order four.

2. Given the initial value problem y′ = −y + t, y(0) = 1. (a) Use the modified Euler method with step size h = 0.2 to compute w_{1}.

(b) Use the Adams-Bashforth 2-step method with the w_{1} of part (a) and step size h = 0.2 to approximate y(0.4).

Continuing problem 2.

(c) Repeat parts (a) and (b) to approximate y(0.4), but this time using step size h = 0.1; i.e., use the modified Euler method to compute w_{1} and then use the Adams-Bashforth 2-step method to approximate y(0.4).

Continuing problem 2.

(d) The actual solution of the initial value problem is y(t) = t − 1 + 2e^{−t}. Use this information to estimate the error in the approximations of y(0.4) in parts (b) and (c). By what factor did the error improve using h = 0.1 instead of h = 0.2? Was that close to what you expected? Explain.

3. Given the initial value problem y′ = e^{y}, y(0) = 0.

(a) Use the Taylor method of order 4 with h = 0.1 to approximate y(0.2).

(b) The actual solution of the initial value problem is y(t) = − ln(1 − t). If you did part (a) correctly your answer should differ from the actual value by less than 0.0000062. Check that it does.

4. Given the initial value problem y′ = y^{2} − t, y(0) = 1

(a) Use the 2-step Adams-Bashforth-Moulton predictor-corrector method to approximate y(0.2) Use a step size of h = 0.1, and use the Taylor method of order 2 to find w_{1}.

(b) Use the standard 4-step Runge-Kutta method to approximate y(0.2).