NOTE: Thus far we have assumed thatp(x)andq(x)iny′+ p(x)y = q(x)are continuous, yet in app...

NOTE: Thus far we have assumed thatp(x)andq(x)iny′+ p(x)y = q(x)are continuous, yet in applications that may not be the case. In particular, the “input”q(x)may be discontinuous. In Example 1, for instance,E(t)inL di/dt + Ri = E(t)may well be discontinuous, such as

We state that in such cases, whereE(t)has one or more jump discontinuities, the solution (11) [more generally, (24) in Section 2.2] is still valid, and can be used in these exercises.

(Newton ’s law of cooling)Suppose that a body initially at a uniform temperatureuois exposed to a surrounding environment that is at a lower temperatureU.Then the outer portion of the body will cool relative to its interior, and this temperature differential within the body will cause heat to flow from the interior to the surface. If the body is a sufficiently good conductor of heat so that the heat transfer within the body is much more rapid than the rate of heat loss to the environment at the outer surface, then it can be assumed, as an approximation, that heat transfer will be so rapid that the interior temperature will adjust to the surface temperature instantaneously, and the body will be at a uniform temperatureu(t)at each instantt.Newton’s law of coolingstates that the time rate of change ofu(t)will be proportional to the instantaneous temperature differenceU−u,so that

where k is a constant.

An interesting application of (15.1) occurs in connection with the determination of the time of death in a homicide. Suppose that a body is discovered at a timeTafter death and its temperature is measured to be 90° F. We wish to solve forT.Suppose that the ambient temperature isU= 70° F and assume-thatu0=98.6° F. Putting this information into the solution to (15.1) we can solve for T, provided that we knowk, but we don’t. Proceeding indirectly, we can infer the value ofkby taking one more temperature reading. Thus, suppose that we wait an hour and again measure the temperature of the body, and find thatu(T +1) = 87° F. Use this information to solve forT.