Stefan’s Law Again An interesting analysis results from playing with the equation of Stefa...

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.