Chapter 3, Problem 10RE

According to Stefan’s law of radiation the absolute temperature T of a body cooling in a medium at constant absolute temperature T_m is given by

$\frac{dT}{dt}=k\left(T^4-T_m^4\right)$,

where k is a constant. Stefan’s law can be used over a greater temperature range than Newton’s law of cooling.

(a) Solve the differential equation.

(b) Show that when T − T_m is small in comparison to T_m then Newton's law of cooling approximates Stefan’s law. [Hint: Think binomial series of the right-hand side of the DE.]