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Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
Download (direct link): elementarydifferentialequations2001.pdf
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When considering the cooling of an egg, ODE (1) is separable because Tout is constant in this instance. Lets consider what happens when the outside temperature changes with time.
We can still use Newtons law of cooling, so that if T(t) is the eggs temperature and Tout(t) is the rooms temperature, then
~77 k( T0ut( t) T) dt
Every ODE text discusses integrating factors and first-order linear ODEs.
Note that ODE (4) is not separable (because Tout varies with time) but it is linear, so we can find its general solution as follows. Rearrange the terms to give the linear ODE in standard form:
77 + kT = Tout{ t) dt
Multiply both sides by ekt, so that
^{^Tt+kT)=kTout(t)^ (5)
Since the left-hand side of ODE (5) is (d/dt)(ektT(t)), it can be rewritten:
jt{e?trI)=kToat(t)eP (6)
Integrating both sides of ODE (6) we have that
ektT = j kToUt(t)ektdt + C
where C is an arbitrary constant. The magic factor fz(t) = ekt that enabled us to do this is called an integrating factor. So ODE (4) has the general solution
T(t) = e~*(J kT0Ut(t)ektdt + Cj
Air Conditioning a Room
Time (minutes)
Figure 3.2: Eggs at initial temperatures of 180, 150, 120, and 90F cool in a room whose temperature oscillates sinusoidally about 70F for k = 0.03 min-1. Do the initial temperatures matter in the long term?
Finally, letting T(0) = T0 and integrating from 0 to t, we get the solution
T(t) = e~kt^j^ kToUt(s)eksds + T^ (7)
It may be possible to evaluate the integral (7) analytically, but it is easier to use ODE Architect right from the start. See Figure 3.2 for egg temperatures in a room whose temperature oscillates between hot and cold.
? Show that if Tout is a constant, then formula (7) reduces to formula (3).
You may find a computer algebra system or a table of integrals helpful!
? Use equation (7) to find a formula for T(t) if
' 2n(t + 3)
ToUt(t) = 82 - 10sin
Use a table of integrals to carry out the integration.
Air Conditioning a Room
Now lets build a model that describes a room cooled by an air conditioner. Without air conditioning, we can model the change in temperature using ODE (1). When the air conditioner is running, its coils remove heat energy at a rate proportional to the difference between Tr, the room temperature, and the
Chapter 3
Newtons law of cooling (twice)!
Time t is measured in minutes.
The modeling here is more advanced than you have seen up to this point. You may want to just use the equations and skim the modeling.
The step function is one of the engineering functions. You can find them by going to ODE Architect and clicking on Help, Topic Search, and Engineering Functions.
temperature Tac of the coils. So, using Newtons law of cooling for the temperature change due to both the air outside the room and the air conditioner coils, our model ODE is

77 = k{ Tout Tr) + kac( Tac Tr) dt
where Tout is the temperature of the outside air and k and kac are the appropriate cooling coefficients. If the unit is turned off, then kac = 0 and this equation reduces to ODE (1).
Lets assume that the initial temperature of the room is 60F and the outside temperature is a constant 100F. The air conditioner operates with a coil temperature of 40F, switches on when the room reaches 80F, and switches off at 70F. Initially, the unit is off and the change in the room temperature is modeled by
= 0.03(100- Tr), Tr(0) = 60 (8)
where we have taken the cooling coefficient k = 0.03 min-1. As we expect, the temperature in the room will rise as time passes. At some time ton the rooms temperature will reach 80F and the air conditioner will switch on. If kac = 0.1 min-1, then for t > ton the temperature is modeled by the IVP
^ = 0.03(100- Tr) + 0.1(40- Tr), Tr(ton) = 80 (9)
which is valid until the room cools to 70F at some time toff. Then for t> toff the room temperature satisfies the IVP (8) but with the new initial condition Tr (toff) = 70. Each time the unit turns on or off the ODE alternates between the two forms given in (8) and (9).
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