Suppose that the ambient temperature of Exercise 42varies sinu soidally with time, as in (...

Suppose that the ambient temperature of Exercise 42varies sinu soidally with time, as in

(4.45)

(a) Find a solution Th of the homogeneous equation T' + kT = 0.

(b) The equation (4.45) is not autonomous, so finding a particular solution Tp is a bit more difficult. However, it doesn’t hurt to guess.4 As a first guess, substitute Tp = C cos ωt + D sin ωt into the equationT' + kT = kA sin cot and show that

(c) Solve the simultaneous equations in part (b) and use Theorem 4.41 to show that the general solution of equation (4.45) is

where F is an arbitrary constant.