1.

The differential equation + y = 0 is to be integrated numerically using the modified Euler method. Use a step size h = 0.1 and let y _n = 1. (a) Solve for the actual numerical error after one step. Compare this result with the value of E _{n+ 1} obtained from (6.86). (b) Using the mean value theorem to obtain E _{n+ 1}, solve for the value of ?, where 0 < ? < 0.1 and t _n =0.

2.

Given a second-order differential equation of the form = f ( y, t). Suppose it is integrated using the algorithm

(a) Find the first error term, (b) Consider the differential equation + y = 0, where ? ₀ is a positive constant. The given algorithm is used to integrate this equation. Analyze the numerical stability by obtaining the corresponding characteristic equation. Determine the range of step size h for numerical instability.

3.

An extrapolation formula

is used to estimate the velocity at time . The step size h is uniform and the values of y _n, y _{n 1} and y _{n 2} are assumed to be known. (a) Solve for the coefficients a, b, and c. (b) Find the first error term.

4.

Given the system

Suppose that Euler integration is used to obtain a solution, (a) Obtain the characteristic...