Ordinary Differential Equations Everything is ordinary about them

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Ordinary Differential EquationsEverything is
ordinary about them
2
Popping tags means

1. Popping bubble wrap
2. Using firecrackers
3. Changing tags of regular items in a store with
   tags from clearance items
4. Taking illicit drugs

3
Physical Examples
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How long will it take to cool the trunnion?
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END
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What did I learn in the ODE class?
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In the differential equation
the variable x is the variable

1. Independent
2. Dependent

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In the differential equation
the variable y is the variable

1. Independent
2. Dependent

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Ordinary differential equations can have these
many dependent variables.

1. one
2. two
3. any positive integer

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Ordinary differential equations can have these
many independent variables.

1. one
2. two
3. any positive integer

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A differential equation is considered to be
ordinary if it has

1. one dependent variable
2. more than one dependent variable
3. one independent variable
4. more than one independent variable

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Classify the differential equation

1. linear
2. nonlinear
3. undeterminable to be linear or nonlinear

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Classify the differential equation

1. linear
2. nonlinear
3. linear with fixed constants
4. undeterminable to be linear or nonlinear

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Classify the differential equation

1. linear
2. nonlinear
3. linear with fixed constants
4. undeterminable to be linear or nonlinear

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The velocity of a body is given by
Then the distance covered by the body from t0 to
t10 can be calculated by solving the
differential equation for x(10) for

1. .

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The form of the exact solution to
is

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END
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Eulers Method
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Eulers method of solving ordinary differential
equations
states

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To solve the ordinary differential equation
by Eulers method, you need to rewrite the
equation as

1. .

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The order of accuracy for a single step in
Eulers method is

1. O(h)
2. O(h2)
3. O(h3)
4. O(h4)

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

1. O(h)
2. O(h2)
3. O(h3)
4. O(h4)

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END
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Do you know how Runge- Kutta 4th Order Method
works?

1. Yes
2. No
3. Maybe
4. I take the 5th

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RUNGE-KUTTA 4TH ORDER METHOD
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Runge-Kutta 4th Order Method
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END
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Physical Examples
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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)?

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Assumptions

• The trunnion is a lumped mass system.
• 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.
• This allows us to make the assumption that the
  temperature is only a function of time and not of
  the location in the trunnion.

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Energy Conservation

• Heat In Heat Lost Heat Stored

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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

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Heat Stored
Heat stored by mass mCT where m mass of
ball, kg C specific heat of the ball, J/(kg-K)
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Energy Conservation
Rate 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
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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) 27oC Temperature of dry-ice/alcohol, Ta
-78oC
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The Differential Equation
Surface area of the trunnion A 2?rL2?r2
2?0.30.6252?0.32
1.744 m2 Mass of the
trunnion M ? V ? (?r2L)
(7800)?(0.3)20.625
1378 kg
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The Differential Equation
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Solution
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
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END
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If assigned HW every class for a grade, you
predict that you would get a

1. better overall grade
2. same overall grade (would not make a difference)
3. lower overall grade

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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)

1. In-class
2. Online

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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)

1. In-class
2. Online

43
How likely are you to watch the YouTube videos
for the topics that were presented in the class
you missed?

1. Certainly
2. Likely
3. Not likely
4. Not at all

44
How likely are you to watch the YouTube videos
for the topics that were presented in the class
you attended?

1. Certainly
2. Likely
3. Not likely
4. Not at all

45
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?

1. Certainly
2. Likely
3. Not likely
4. Not at all

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Given
The value of
at y(4) using finite difference method and a
step size of h4 can be approximated by

1. .