Consider anew Newton’s law of cooling, where the temperature of a body is modeled by the e...

Consider anew Newton’s law of cooling, where the temperature of a body is modeled by the equation

(4.43)

where T is the temperature of the body and A is the temperature of the surrounding medium (ambient temperature). Although this equation is linear and its solution can be found by using an integrating factor, Theorem 4.41 provides a far simpler approach.

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

(b) Find a particular solution Tp of the inhomogeneous equation (4.43). Note: This equation is autonomous. See Section 2.9.

(c) Form the general solution T = Th + Tp.

(d) Add a source of constant heat (like a heater in a room) to the model (4.43), as in (4.44)

Use the technique outlined in parts (a)–(c) to find the general solution of equation (4.44).