block 5 stochastic & dynamic systems lesson 18 – differential equations - methods and models...
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Block 5 Stochastic & Dynamic Systems Lesson 18 – Differential Equations - Methods and Models
Dynamic Systems
Charles EbelingUniversity of Dayton1
Differential Equations
Equations involving derivatives used to solve practical problems in engineering and science.
ENM/MSC applications include: queuing analysis reliability Markov processes modeling population growth, decline, spread
of disease modeling conflicts
2
Examples of Differential Equations
2
2
3 2
3 2
2 2
2 2
22
2
32
sin cos
0
( 1) 0
d s
dt
d y d yx y x
dx dx
u u
x y
d y dyx x xy
dx dx
I get it now. A differential
equation has those dy-dx things in it.
3
Ordinary Differential Equations
Single independent variable with no partial derivatives
order – highest ordered derivative degree – algebraic degree of the highest ordered
derivative solution – a non-derivative relationship between the
variables which satisfy the diff. eq. as an identify Examples:2
2 22
2 33 22
3 2
2 2
2 0
d y dyx x y xdx dx
d y d y dy dyx
dx dx dx dx
order 2, degree 1
order 3, degree 2
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Solutions General solution – solutions involving n constants for
an nth order differential equations as a result of n integrations
Particular solution – obtained from general solution by assigning specific values to the constants
Example – show y = cx2 is a general solution to:
2
2
2
since 2
then 2 2 2
dyx ydx
d cxcx
dxdyx x cx cx ydx
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Our very next diff. equation
Example – show
is a general solution to:
2 2x xy Ae Be x x
22
2
2
22
2
2 2
?2 2 2
2 2
2 3 2
2 2 1
4 2
4 2 2 2 1
2 3 2
3 2 3 2
x x
x x
x x x x
x x
d y dyy x
dx dxdy
Ae Be xdx
d yAe Be
dx
Ae Be Ae Be x
Ae Be x x x
x x
Who would have thought? It isn’t so bad when you are given the answer.
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First Order – Linear (1st degree)
( , ) 0
( , ) ( , ) 0
dyF x y
dxM x y dx N x y dy
7
Solving first-order, linear equations
Variables separable Homogeneous equations Exact Equations Linear in y
Engineering ManagementStudents enjoying solving adifferential equation problem. 8
Variables Separable - Examples
( ) ( ) 0
solve by direct integration
( ) ( )
A x dx B y dy
A x dx B y dy C
First he says its
separable then he says to integrate. I am totally confused.
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A Separable Example
2
2
2
2
solve: 5
rewrite: 5
05
now integrate:5
15 5 ln( 5)
1or
5 ln( 5)
dxxy xdy
xy dx x dy
x dydx
x y
x dydx C
x y
x x Cy
yc x x
from tables
10
An old friend
A = the amount of an initial investment at an interest rate of i compounded continuously
0 0
ln ln
ln ln ln
at 0, and
it it
it
dAiA
dtdA
i dt CAA it C
AA C it
CA
e or A CeC
t A A A A e
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Another Example
2
2
3
3
3
3
0
0
0
3
3
x y y
yx
y
x y
x y
yx
x y
dxe e e
dy
ee dx dy
e
e dx e dy
e dx e dy C
ee C
e e C
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1st Order Linear Diff Equations – General Solution
( )
( ) ( )
is an integrating factor
( ) ( )
( )
P x dx
P x dx P x dx
dyP x y Q x
dx
e
ye e Q x dx C
An integrating factor? Why it is a
factor which when multiplied through a differential equation results in an exact differential equation. But everyone knows
that. 13
Example #1
2 2
2 2
2
2
2
( ) 2; ( )
integrating factor:
y
y
x
x
dx x
x x x x
x x
x x
dyy e
dx
P x Q x e
e e
e e e dx e dx C
e e C
y e Ce
( ) ( )( )
P x dx P x dxye e Q x dx C
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Check your work!
2
2
2 2
2
soln?
2
2 2 2
x
x x
x x
x x x x x
dyy e
dx
y e Ce
dye Ce
dxdy
y e Ce e Ce edx
Not checking your work is indeed
dumb.15
Example #2
2/ 2 ln
5
4
2
4
2 2
32
2
5 2
rewrite:
integrating factor:
2
2
1
31
3
x dx xe e
dyx y xdx
dyy x
dx x
x
y xdx C
x x
y xx dx C C
x
y x Cx
( ) ( )( )
P x dx P x dxye e Q x dx C
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Check
5
5 2
4
4 5 2 5
2
1
35
23
5 12 2
3 3
dyx y xdx
y x Cx
dyx Cx
dx
x x Cx x Cx x
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Dynamic Models
Putting your math to work…
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Exponential GrowthA population increases at a rate proportional to its size. If a population doubles in one year, how many years before it will be 1000 times its original size? Let N = population size at time t and N0 = population at t = 0
ln
'kt C C kt kt
dNkN
dtdN
kdtNdN
kdt CNN kt C
N e e e C e
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More Population Growth
(0)0 0
(1)0 0
0
.69314720 0
' , therefore '
2 and ln 2 ; .6931472
for 1000 :
1000
ln10009.9657 yrs
.6931472
k
k
t
N C e C N
N N e k k
N N
N N e
t
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The Logistics Curve
Under exponential growth, a population gets infinitely large as time goes on
In realty, there are limits to growth that will slow down the rate of growth
food supply, predators, overcrowding, disease, etc. Assume the size of a population, N, is limited
to some maximum number M, where 0 < N < M and as NM, dN/dt 0
Desire exponential growth initially but then limits to growth
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The Model
dN M NkN
dt M
For N small, (M-N) / M is close to 1 and growth is approximately exponential
Then as NM, (M-N) 0 and dN/dt 0
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Let’s solve for N…
1
ln
ln
dNKN M N
dtdN
KdtN M N
dNKdt C
N M N
NKt C
M M N
NMKt MC
M N
1ln
dx x
x a bx a a bx
from the old table of integrals:
23
Keep on solving…
ln
1
11 11
MKt MC MKt MC MKt
MKt
MKt MKt
MKt
MKt ctMKt
NMKt MC
M N
Ne e e Ae
M N
N M N Ae
N Ae MAe
MAe M MN
Ae beeA
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The Logistics Curve
.2
100
1 1 10ct t
MN
be e
Logistic Curve
0
20
40
60
80
100
120
0 10 20 30 40 50 60
t
N
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The Maximum Rate of Growth
2
2 0
*2
dNKN M N
dt
d KNM KNd KN M NK M N
dN dNM
N
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Now find the time at which the maximum growth rate occurs…
1
2 1
1 2
1 2
1
ln
ct
ct
ct
ct
ct ct
MN
beM M
be
be M M
be
e or e bbb
tc
substitute M/2 for N
solve for t
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Time for an example
The new Montgomery County jail can hold a maximum of 800 prisoners. One year ago there were 50 prisoners in the jail and now there are 200. Assuming the jail population follows a logistic function, how many prisoners will there be three years from now?Let N(t) = the jail population after t years since openingwhen t = 0:
( )1
80050
115
ct
MN t
be
bb
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Still time for an example
800( )
1 15When 1 200
800200
1 15800
1 15 4200
3 1ln 5
15 5
ct
c
c
c
N te
t N
e
e
e or c
3
800( )
11 15
5
800(3) 781
11 15
5
tN t
N
I’ll be out of this joint
before then.
Max growth rate occurs at t = ln(15)/ln5 =1.7 years29
A Campus Rumor
In a Midwestern university, having a student population of 10,200, a rumor had been initiated among a class of 30 students that a favorite engineering professor will be winning the Noble prize in economics for his work on multivariable nonlinear profit maximization based upon first and second order partial derivatives. After a week, the rumor had spread around campus to 160 students. When the semester ends 5 weeks later (week 6 since rumor initiation), how many students will know the rumor? Assume the logistics growth process.
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The rumor is spreading…
( )1
10, 20030
1339
ct
MN t
be
bb
10, 200( )
1 339When 1 160
10, 200160
1 33910, 200
1 339 63.75160
62.75.1851 ln .1851
339
ct
c
c
c
N te
t N
e
e
e or c
31
still spreading…
6
10, 200( )
1 339 .1859
10, 200(6) 10, 059
1 339 .1859
tN t
N
Yes, it’s all over the campus. He has
won the Noble prize in economics. I am so glad I took his
course.
32
All good things must come to an end.
An engineering professor caught grading student
papers
Hey, don’t run away from solving these equations
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ENM 503 students saddened as they approach the end of the course
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