Use Euler’s method with the spacings h = 0.5, 0.1, 0.05, 0.01 to approximate the solutio...

Use Euler’s method with the spacings h = 0.5, 0.1, 0.05, 0.01 to approximate the solution to the initial value problem

on the interval (The explanation for the erratic results lies in Problem 18 of Exercises 1.2.)

Heat Exchange. There are basically two mechanisms by which a physical body exchanges heat with its environment. The contact heat transfer across the body’s surface is driven by the difference in the body’s temperature and that of the environment; this is known as Newton’s law of cooling. However, heat transfer also occurs due to thermal radiation, which according to Stefan’s law of radiation is governed by the difference of the fourth powers of these temperatures. In most cases one of these modes dominates the other. Problems 15 and 16 invite you to simulate each mode numerically for a given set of initial conditions.