The differential equation where k1 and A0 are positive constants, can be used t...

The differential equation

where k₁ and A₀ are positive constants, can be used to model the temperature variation T (t) in a building. In this equation, the first term on the right-hand side gives the contribution due to the variation in the outside temperature, and the second term on the right-hand side gives the contribution due to the heating effect from internal sources such as machinery, lighting, people, and so on. Consider the case when where A and B are constants, and t is measured in hours.

(a) Make a sketch of T_m(t). Taking t = 0 to correspond to midnight, describe the variation of the external temperature over a 24-hour period.

(b) With T_m given in (1.6.14), solve (1.6.13) subject to the initial condition T (0) = T₀.