Transcript Sec 1.3 Differential Equations as Mathematical Models Sec 3.1 Linear Model

Sec 1.3 Differential Equations as
Mathematical Models
Sec 3.1 Linear Model
Where it appear
Mechanical Eng.
x' '2x' x  f (t )
2
Dr. Faisal fairag
Chemical Eng.
dA
 ( Input rate of salt) - ( Output rate of salt)
dt
dA
 Rin  Rout
dt
Dr. Faisal fairag
Electrical Eng.
2
d q
dq 1
L 2 R
 q  E (t )
dt
dt C
Dr. Faisal fairag
Civil Eng.
2
d y
dy 2
 (w / T ) 1  ( )  0
2
dx
dx
Dr. Faisal fairag
Heat Transfer
dT
 k (T  Tm )
dt
Dr. Faisal fairag
Civil Eng.
2
d x
m 2  F ( x)  g (t )
dt
Dr. Faisal fairag
dv
m
 mg  kv
dt
Dr. Faisal fairag
Sec 1.3 Differential Equations as
Mathematical Models
Sec 3.1 Linear Model
Behavior
of some real-life system
or phenomenon
Mathematical Model
Construction of a Mathematical model:
STEP 1:
STEP 2:
Identification of the variables
We make some reasonable assumptions (physical laws)
Assumptions
Express DE
Mathematical
Formulation
Solve DE
If necessary alter
assumptions
Check model
Predictions with
Known facts
Display graphically
Obtain
Solutions
Applications
1.Population Dynamics*
2.Radioactive Decay*
3.Spread of Disease*
4.Chemical Reactions
5.Newton’s Law of Cooling*
6.Mixtures*
7.Draining a Tank
8.Series Circuits
9.Falling Bodies
10.Falling Bodies and Air Resistance
11.Suspended Cables
EXAMPLE:
When a pizza is removed from an oven. Its temperature is measured at 300 F.
Three minutes later its temperature is 200 F. How long will it take for the pizza
to cool off to a room temperature of 70 F?
Newton’s Law of Cooling-Warming: the rate at which the tempof a body changes
is proportional to the difference between the temp of the body and the temp of
the surrounding.
t  time
T(t)  temp of the pizza at time t
dT
 the at which t he temp of the pizza changes
dt
dT

dt
(T  Tm )
EXAMPLE:
When a pizza is removed from an oven. Its temperature is measured at 300 F.
Three minutes later its temperature is 200 F. How long will it take for the pizza
to cool off to a room temperature of 70 F?
Newton’s Law of Cooling-Warming: the rate at which the tempof a body changes
is proportional to the difference between the temp of the body and the temp of
the surrounding.
230 exp(-0.19018 t)+70
240
220
200
180
160
140
120
100
80
0
5
10
15
20
25
time in minutes
30
35
40
#14/p99:
A thermometer is taken from a inside room to the outside, where the air
temperature is 5F. After I minute the thermometer reads 55F, and after 5
minutes it reads 30F. What is the initial temperature of the inside room?
Newton’s Law of Cooling-Warming: the rate at which the temp of a body changes
is proportional to the difference between the temp of the body and the temp of
the surrounding.
#1/p99:
The population of a community is known to increase at a rate proportional to
the number of people present at time t. If an initial population P_0 has doubled
in 5 years, how long will it take to triple? Quadruple?
Population: The population of a community is known to
increase at a rate proportional to the number of people present at time t
Population: The population of a community is known to
increase at a rate proportional to the number of people present at time t
2002
0
23,500,000
2003
1
24,250,000
P(t )  23,500,000e0.0314t
2007  t  5  P(5)  27,495,000
c  23,500,000
2425
k  ln
 0.0314
2350
Interested:
See #39/101
Read sec 3.2
A large tank held 300 gallons of brine solution.
Salt was entering and leaving the tank; a brine
solution was being pumped into the tank at the
rate of 3 gal/min. it mixed with the solution there,
and then the mixture was pumped out at the rate
of 3 gal/min. The concentration of the salt in the
inflow was 2 lb/gal. If 50 pounds of salt were
dissolved initially in the 300 gallons, how much
salt is in the tank after 10 min?
input rate
3 gal/min
A(t )  amount of salt at time t
dA
 the rate at which A(t) changes
dt
300
gallons
 input rate 
 output rate 
dA
  

 
dt
 of salt 
 of salt 

R in

R out
Rin  input rate at which salt enters
Rout  output rate at which salt leaves
Output rate
3 gal/min
A large tank held 300 gallons of brine solution. Salt was entering and
leaving the tank; a brine solution was being pumped into the tank at the
rate of 3 gal/min. it mixed with the solution there, and then the mixture
was pumped out at the rate of 3 gal/min. The concentration of the salt in
the inflow was 2 lb/gal. If 50 pounds of salt were dissolved initially in the
300 gallons, how much salt is in the tank after 10 min?
dA

dt
R in

R out
2 lb/gal 
3 gal/min
R out
 concentrat ion 
 input rate 




R in  
of salt
   of

 inflow

 brine 




R in 
input rate

3 gal/min 
300
gallons
 6 lb/min
 concentrat ion 
 output rate 





of salt
of
  

 outflow 
 brine 




R in 
? lb/gal 

3 gal/min 
Output rate
3 gal/min
A large tank held 300 gallons of brine solution. Salt was entering and
leaving the tank; a brine solution was being pumped into the tank at the
rate of 3 gal/min. it mixed with the solution there, and then the mixture
was pumped out at the rate of 3 gal/min. The concentration of the salt in
the inflow was 2 lb/gal. If 50 pounds of salt were dissolved initially in the
300 gallons, how much salt is in the tank after 10 min? how much salt in
the tank after a long time?
dA
A
 6 
dt
100
Sol:
t
100
3 gal/min
A(0)  50
A(0)  50  C  550
A(t)  600  Ce
A(t)  600  550e
input rate
t
100
A( 10 )  600  550e
 102.3394
300
gallons














10
100
600
550
500
450
Amount
400
of
Salt
350
A
300
250
200
150
Output rate
100
0
100
200
300
Time (minutes)
400
500
600
3 gal/min
#19
p99
A tank held 200 liters of fluid in which 30 grams of salt is dissolved.
Brine containing 1 gram of slat per liter is then pumped into the tank at a
rate of 4 L/min; the well-mixed solution is pumped out at the same rate.
Find the number A(t) of grams of salt in the tank at time t.
input rate
4 L/min
200
liters
Output rate
4 L/min
Chapter-Summary
Next week (mond –wed)
Chapter # 1 and
Chapter # 2
Chapter
Date of
submission
GROUP 1
1)AL-GHAMDI, MOHAMMAD TURKI
2)AL-ZAHRANI, AHMAD ALI AHMAD
3)L-HADI, MOHAMMAD SALEH HADAJ
4)AL-MUTAIRI, ABDUL-LATIF THUWAINI
1
Mon 2/25/2008
11:58 AM
GROUP 2
1)Khalid Abdulghani
2)Khaldoon Al-Azzah
3)Yousef Al-Shaheen
4)Abdulrahman Al-Saggaf
5)Mohammad Bawazeer
6)Ibraheem Alsufyani
2
Mon 2/25/2008
12:24 PM
GROUP
Members
Expected Date
for presentation