In many of the following problems, it will be essential to have a calculator or comp...

In many of the following problems, it will be essential to have a calculator or computer available. You may use a software package† or write a program for solving initial value problems using the improved Euler’s method algorithms on pages 127 and 128. (Remember, all trigonometric calculations are done in radians.)

Falling Body. In Example 1 of Section 3.4, we modeled the velocity of a falling body by the initial value problem under the assumption that the force due to air resistance is -bv. However, in certain cases the force due to air resistance behaves more like -bv^r where r is some constant. This leads to the model

To study the effect of changing the parameter r in (14), take m = 1, g = 9.81, b = 2, and v0 = 0. Then use the improved Euler’s method subroutine with h = 0.2 to approximate the solution to (14) on the interval What is the relationship between these solutions and the constant solution v(t) = (9.81/2)^1/r.