Chapter4
Chapter 4
Applications of First-order Differential Equations to Real World Systems
4.1 Cooling/Warming Law
4.2 Population Growth and Decay
4.3 Radio-Active Decay and Carbon Dating
4.4 Mixture of Two Salt Solutions
4.5 Series Circuits
4.6 Survivability with AIDS
4.7 Draining a tank
4.8 Economics and Finance
4.9 Mathematics Police Women
4.10 Drug Distribution in Human Body
4.11 A Pursuit Problem
4.12 Harvesting of Renewable Natural Resources
4.13 Exercises
In Section 1.4 we have seen that real world problems can be represented by first-order differential equations.
In chapter 2 we have discussed few methods to solve first order differential equations. We solve in this chapter first-order differential equations modeling phenomena of cooling, population growth, radioactive decay, mixture of salt solutions, series circuits, survivability with AIDS, draining a tank, economics and finance, drug distribution, pursuit problem and harvesting of renewable natural resources.
4.1 Cooling/Warming law
We have seen in Section 1.4 that the mathematical formulation of Newton’s empirical law of cooling of an object in given by the linear first-order differential equation (1.17)
This is a separable differential equation. We have
or ln|T-Tm |=t+c1 or T(t) = Tm+c2et (4.1)
Example 4.1: When a chicken is removed from an oven, its temperature is measured at 3000F. Three minutes later its temperature is 200o F. How long will it take for the chicken to cool off to a room temperature of 70oF.
Solution: In (4.1) we put Tm = 70 and T=300 at for t=0.
T(0)=300=70+c2e.0
This gives c2=230
For t=3, T(3)=200
Now we put t=3, T(3)=200 and c2=230 in (4.1) then
200=70 + 230 e.3 or or or Thus T(t)=70+230 e-0.19018t (4.2)
We observe that (4.2) furnishes no finite solution to T(t)=70 since limit T(t) =70. t ¥
The temperature variation is shown graphically in Figure 4.1. We observe that the limiting temperature is 700F.
Figure 4.1
4.2 Population Growth and Decay
We
Applications of First-order Differential Equations to Real World Systems
4.1 Cooling/Warming Law
4.2 Population Growth and Decay
4.3 Radio-Active Decay and Carbon Dating
4.4 Mixture of Two Salt Solutions
4.5 Series Circuits
4.6 Survivability with AIDS
4.7 Draining a tank
4.8 Economics and Finance
4.9 Mathematics Police Women
4.10 Drug Distribution in Human Body
4.11 A Pursuit Problem
4.12 Harvesting of Renewable Natural Resources
4.13 Exercises
In Section 1.4 we have seen that real world problems can be represented by first-order differential equations.
In chapter 2 we have discussed few methods to solve first order differential equations. We solve in this chapter first-order differential equations modeling phenomena of cooling, population growth, radioactive decay, mixture of salt solutions, series circuits, survivability with AIDS, draining a tank, economics and finance, drug distribution, pursuit problem and harvesting of renewable natural resources.
4.1 Cooling/Warming law
We have seen in Section 1.4 that the mathematical formulation of Newton’s empirical law of cooling of an object in given by the linear first-order differential equation (1.17)
This is a separable differential equation. We have
or ln|T-Tm |=t+c1 or T(t) = Tm+c2et (4.1)
Example 4.1: When a chicken is removed from an oven, its temperature is measured at 3000F. Three minutes later its temperature is 200o F. How long will it take for the chicken to cool off to a room temperature of 70oF.
Solution: In (4.1) we put Tm = 70 and T=300 at for t=0.
T(0)=300=70+c2e.0
This gives c2=230
For t=3, T(3)=200
Now we put t=3, T(3)=200 and c2=230 in (4.1) then
200=70 + 230 e.3 or or or Thus T(t)=70+230 e-0.19018t (4.2)
We observe that (4.2) furnishes no finite solution to T(t)=70 since limit T(t) =70. t ¥
The temperature variation is shown graphically in Figure 4.1. We observe that the limiting temperature is 700F.
Figure 4.1
4.2 Population Growth and Decay
We