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

4

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

5

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.

6

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.

9

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

11

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

12

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

14

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

16

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

17

Dynamic Models

Putting your math to work…

18

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

19

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

20

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

21

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

22

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

24

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

25

The Maximum Rate of Growth

2

2 0

*2

dNKN M N

dt

d KNM KNd KN M NK M N

dN dNM

N

26

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

27

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

28

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.

30

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

33

ENM 503 students saddened as they approach the end of the course

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