Stefan’s Law Again According to Stefan’s Law of Radiation (previously examined in Sec. 1.3...

Stefan’s Law Again According to Stefan’s Law of Radiation (previously examined in Sec. 1.3, Problem 55 and Sec. 1.4, Problem 11), the rate of change of the radiation energy of a body at absolute temperature T is given by dT/dt = k(M4 − T4), where k > 0 and M is the ambient or surrounding absolute temperature. Sketch typical solutions T = T(t) for various initial temperatures T0 = T(0).

Sec. 1.3, Problem 55

Calculator or Computer With the help of suitable computer software, for Problem graph the families of curves along with their families of orthogonal trajectories.

x2 + y2 = c (coaxial circles)

Sec. 1.4, Problem 11

Stefan’s Law Again An interesting analysis results from playing with the equation of Stefan’s Law (Sec. 1.3, Problem 59). For dT/dt = k(M4 − T4), let k = 0.05, M = 3, T(0) = 4.

(a) Estimate T(1) by Euler’s method with step sizes h = 0.25, h = 0.1.

(b) Graph a direction field and both multistep approximations from (a). Explain why and how the approximations from (a) take different routes.

(c) Find an equilibrium solution; relate it to (a) and (b).

Sec. 1.3, Problem 59

Radiant Energy Stefan’s Law of Radiation states that the radiation energy of a body is proportional to the fourth power of the absolute temperature T of a body.6 The rate of change of this energy in a surrounding medium of absolute temperature M is thus

where k > 0 is a constant. Show that the general solution of Stefan’s equation is

where c is an arbitrary constant.