Ordinary Differential Equations Everything is ordinary about them.

Ordinary Differential EquationsEverything is ordinary about them

Popping tags means

A. Popping bubble wrap

B. Using firecrackersC. Changing tags of

regular items in a store with tags from clearance items

D. Taking illicit drugs

Popping bubble w

rap

Using firecra

ckers

Changing tags o

f regu

lar ...

Taking illici

t dru

gs

25% 25%25%25%

Physical Examples

How long will it take to cool the trunnion?

),( ahAdt

dmc

room )0(

What did I learn in the ODE class?

In the differential equation

Inde

pende

nt

Dep

enden

t

0%0%

6)0(,3 yeydx

dy x

the variable x is the variable

A. IndependentB. Dependent

In the differential equation

Inde

pende

nt

Dep

enden

t

0%0%

6)0(,3 yeydx

dy x

the variable y is the variable

A. IndependentB. Dependent

Ordinary differential equations can have these many dependent variables.

one

two

any

positiv

e in

tege

r

33% 33%33%

A. one B. twoC. any positive integer

Ordinary differential equations can have these many independent variables.

33% 33%33%A. oneB. twoC. any positive integer

A differential equation is considered to be ordinary if it has

0% 0%0%0%

A. one dependent variableB. more than one dependent variableC. one independent variableD. more than one independent variable

Classify the differential equation

33% 33%33%

5)0(,342 yeydx

dy x

A. linearB. nonlinearC. undeterminable to be

linear or nonlinear


0% 0%0%0%

5)0(,32 yexydx

dy x

A. linearB. nonlinearC. linear with fixed constantsD. undeterminable to be

linear or nonlinear


0% 0%0%0%

5)0(,32 2 yeydx

dy x

A. linearB. nonlinearC. linear with fixed constantsD. undeterminable to be

linear or nonlinear

The velocity of a body is given by

A. .

B.

C.

D. .

0% 0%0%0%

0)0(,2 2 xedt

dx t

0)0(,52 xedt

dx t

5)0(,52 xedt

dx t

0)0(,52

2

xte

dt

dx t

0,5)( 2 tetv t

Then the distance covered by the body from t=0 to t=10 can be calculated by solving the differential equation for x(10) for

The form of the exact solution to

1 2 3 4

25% 25%25%25%

xx BeAe 5.1

xx BxeAe 5.1

xx BeAe 5.1

xx BxeAe 5.1

5)0(,32 yeydx

dy x is

1.

2.

3.

4.

Euler’s Method

Euler’s method of solving ordinary differential equations

A. B. C. D.

25% 25%25%25%

hyxfyy iiii ),(1

hyxfyy ii ),(1

),(1 iiii yxfyy

hyxfy iii ),(1

0)0(),,( yyxfdx

dy states

A.

B.

C.

D.

To solve the ordinary differential equation

A. .

B.

C.

D.

.

0% 0%0%0%

5)0(),3

5cos(

3

1 3

yy

xdx

dy

5)0(,5sin 2 yyxdx

dy

5)0(),5(sin3

1 2 yyxdx

dy

5)0(,sin3

1 yx

dx

dy

,5)0(,sin53 2 yxydx

dy

by Euler’s method, you need to rewrite the equation as

The order of accuracy for a single step in Euler’s method is

A. O(h)B. O(h2)C. O(h3)D. O(h4)

O(h)O(h2)

O(h3)O(h4)

25% 25%25%25%

The order of accuracy from initial point to final point while using more than one step in Euler’s method is

A. O(h)B. O(h2)C. O(h3)D. O(h4)

O(h)O(h2)

O(h3)O(h4)

25% 25%25%25%

Do you know how Runge- Kutta 4th Order Method works?A. YesB. NoC. MaybeD. I take the 5th

50%50%

RUNGE-KUTTA 4TH ORDER METHOD

26

Runge-Kutta 4th Order Method

hkkkkyy ii 43211 226

1

ii yxfk ,1

hkyhxfk ii 12 2

1,

2

1

hkyhxfk ii 23 2

1,

2

1

hkyhxfk ii 34 ,

0)0(),,( yyyxfdx

dy

Physical Examples

Ordinary Differential Equations

Problem: The trunnion initially at room temperature is put in a bath of dry-ice/alcohol. How long do I need to keep it in the bath to get maximum contraction (“within reason”)?

AssumptionsThe trunnion is a lumped mass system.

a. What does a lumped system mean? It implies that the internal conduction in the trunnion is large enough that the temperature throughout the ball is uniform.

b. This allows us to make the assumption that the temperature is only a function of time and not of the location in the trunnion.

Energy Conservation

Heat In – Heat Lost = Heat Stored

Heat Lost

Rate of heat lost due to convection= hA(T-Ta)

h = convection coefficient (W/(m2.K)) A = surface area, m2

T= temp of trunnion at a given time, K

Heat Stored

Heat stored by mass = mCT

wherem = mass of ball, kgC = specific heat of the ball, J/(kg-K)

Energy ConservationRate at which heat is gained – Rate at which heat is lost=Rate at which heat is stored

0- hA(T-Ta) = d/dt(mCT)

0- hA(T-Ta) = m C dT/dt

Putting in The Numbers

Length of cylinder = 0.625 m Radius of cylinder = 0.3 m

Density of cylinder material = 7800 kg/m3

Specific heat, C = 450 J/(kg-C)Convection coefficient, h= 90 W/(m2-C)Initial temperature of the trunnion, T(0)= 27oCTemperature of dry-ice/alcohol, Ta = -78oC

The Differential EquationSurface area of the trunnion

A = 2rL+2r2

= 2**0.3*0.625+2**0.32

= 1.744 m2

Mass of the trunnionM = V

= (r2L) = (7800)*[*(0.3)2*0.625]

= 1378 kg

The Differential Equation

dt

dTmCTThA a )(

dt

dTT 4501378)78(744.190

27)0(

)),78(10531.2 4

T

Tdt

dT

Solution

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-80

-60

-40

-20

0

20

40

Time in seconds

Tem

pera

ture

in C

elci

us

Exact and Approximate Solution of the ODE by Euler's Method

Exact

Approximation

Time Temp (s) (oC) 0 27 1000 0.42 2000 -19.42 3000 -34.25 4000 -45.32 5000 -53.59 6000 -59.77 7000 -64.38 8000 -67.83 9000 -70.40 10000 -72.32

If assigned HW every class for a grade, you predict that you would get a

A. B. C.

33% 33%33%

A. better overall gradeB. same overall grade (would

not make a difference)C. lower overall grade

If I had given you a choice of taking the class online or in-class, and class attendance was not mandatory for in-class section, what would have been your choice? (you would have the same graded assignments and had to come to campus to take the tests for either section)

In-c

lass

Onlin

e

50%50%

A. In-classB. Online

If I had given you a choice of taking the class online or in-class but required 80% attendance for in-class section, what would have been your choice? (you would have the same graded assignments and had to come to campus to take the tests for either section)

In-c

lass

Onlin

e

50%50%

A. In-classB. Online

How likely are you to watch the YouTube videos for the topics that were presented in the class you missed?

Certainly

Likely

Not li

kely

Not a

t all

25% 25%25%25%A. CertainlyB. LikelyC. Not likelyD. Not at all

How likely are you to watch the YouTube videos for the topics that were presented in the class you attended?

Certainly

Likely

Not li

kely

Not a

t all


If based on your background such as learning patterns, GPA, etc, you were recommended to register for the online section or in-class section, how likely are you going to accept the recommendation?

Certainly

Likely

Not li

kely

Not a

t all


Given

A. .

B.

C.

D.

,0)12(,0)0(,5.06 22

2

yyxxdx

yd

.

0% 0%0%0%

16

)4()8(2)12( yyy

8

)0()8( yy

16

)0()4(2)8( yyy

4

)0()4( yy

The value of 2

2

dx

yd

at y(4) using finite difference method and a step size of h=4 can be approximated by

Ordinary Differential Equations Everything is ordinary about them.

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