This is a running assignment. Work through these problems as we cover each method.

PROBLEM:  Approximate the solution of the initial value problem (IVP)

y '   =   5 sin(2x)   –   y ,

y(0)   =   –1 ,

from x = 0 to x = 1 with stepsize h = 0.2 using:

1. Euler's method,

2. the implicit Euler method,

3. the trapezoidal method,

4. the modified Euler (predictor-corrector) method,

5. the classical Runge-Kutta method.

For comparison, from MATH-204 we find that the exact solution of the IVP is

y*(x)   =   e^–x + sin(2x) – 2 cos(2x) .

In each method above, determine the error of the approximation at each node. For example, the error at node n is

y_n – y*(x_n) .

NOTE:

• You might use a spreadsheet to calculate the approximate results, the exact solutions, and the errors, and play around with plotting them.

• We also discuss the Runge–Kutta–Fehlberg and the Runge–Kutta–Verner methods, but there is no problem assigned for them. However, you are still responsible for understanding those methods per what we discuss about them in class.

• Refer to the handout Runge–Kutta–Fehlberg Method.

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