Math 274 Differential Equations
Fall 2015
Information
Instructor
Textbook
Differential Equations (8th ed.) Nagle, Saff, Snider
Section
02
Office Hours
Schedule (Wil 2-236)
Phone
994-5356
Classroom
- MF 11:00-11:50am JAB 215
- MF 11:00-11:50am Herrick Hall 117
- TR 11:00-11:50am (ROBH 113)
JAB 215 is a great room.
Final: Thursday, Dec 10 from 10:00-11:50 in ROBH 218
Grading, Syllabus, and Homework
Grading
The course % is determined by:
Midterm 1 M1 100
Midterm 2 M2 100
Final F 100
Quizzes Q 100
________________________________
400
% = (M1+M2+F+HW)/4
- The final is not comprehensive. Six quizzes each worth 20 points will be given. Your best 5 quiz scores determine Q above.
- Exam and quiz dates are indicated on the schedule below. Their content will be announced in class.
- All exams and quizzes are closed book and no electronic devices are permitted. All exams including the Final will be given in class: Wil 1-144
Syllabus
Material covered in text is from:
- Chapter 1 Introductory Definitions
- Chapter 2 First Order ODE Methods
- Chapter 3 First Order Models
- Chapter 4 Second Order Linear ODE Methods
- Chapter 6 Higher Order Differential Equations
- Chapter 7 Laplace Transforms
- Chapter 9 Linear Systems
Homework
- Suggested homework is listed below.
- Although the homework is not graded it is representative of the kinds of questions which will be on quizzes and exams.
- Some additional problem sets and/or handouts will be handed out in class and/or posted on this site below.
Schedule
Below is a calendar showing the schedule of quizzes and tests (red) and holidays (green).
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Sunday
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Monday
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Tuesday
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Wed
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Thursday
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Friday
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Saturday
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23
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24
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25
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26
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27
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28
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29
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30
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31
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1
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2
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3
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4
Quiz 1 |
5
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6
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7
Labor Day |
8
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10
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11
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12
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13
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14
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15
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16
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17
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18
Quiz 2 |
19
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20
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21
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22
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23
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24
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25
Midterm 1 |
26
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27
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28
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29
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30
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1
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2
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3
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4
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5
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6
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7
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8
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9
Quiz 3 |
10
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11
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12
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13
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15
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16
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17
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18
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19
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20
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21
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22
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23
Quiz 4 |
24
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25
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26
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27
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28
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29
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30
Midterm 2 |
31
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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11
Veteran's Day |
12
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13
Quiz 5 |
14
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15
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16
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17
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18
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19
Last Drop Day |
20
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21
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22
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23
Quiz 6 |
24
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25
Thanksgiving |
26
Thanksgiving |
27
Thanksgiving |
28
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29
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30
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1
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2
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3
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4 Classes end
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5
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6
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7
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8
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9
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10
Final ROBH 218 10:00-11:50am |
11
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12
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Suggested Homework and Syllabus
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1.1
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1,2,5,7,9,11
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Dependent/independent variables, linear ODE
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1.2
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1a,2a,3,5,7,9,11,21,23,27,29a
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Solutions, Existence, Initial Value Problem
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1.3
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not covered
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Direction Fields
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1.4
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not covered
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Euler's Method
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2.1
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none
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Motion of a Falling Body
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2.2
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1,2,3,5,7,8,9,11,17,18,19,23
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1rst Order Separable
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2.3
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2,3,4,7,9,10,13,15,17,18,19,22
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1rst Order Linear
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2.4
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1,2, 5 (solve as well),11,12,13,22,25,26
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1rst Order Exact
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2.5
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not covered
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1rst Order Special Integrating Factors
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2.6
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5,7,9,11 (implicit),15,21,23,25
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1rst Order Homogeneous and Bernoulli only
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3.1
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none
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Mathematical Modelling
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3.2
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1,3,7
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Mixing models (only)
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3.3
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1,3,5
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Heating and Cooling Problems
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3.4
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1,5,24(hard)
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Newtonian Mechanics
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3.5
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not covered
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Electrical Circuits
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3.6
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not covered
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Improved Euler Methods
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3.7
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not covered
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Higher Order Numerical Methods
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Midterm 1
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4.1
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none
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Introductory 2nd Order Models
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4.2
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1,5,9,13,19,27,31,37(r=1 root), 39 (r=2), 43
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Homogeneous IVP, existence, Real Roots Case
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4.3
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1,3,5,9,11,13,19(r=1),21,25,29b (r=2),29c
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Homogenous, Complex Roots Case
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4.4
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9,11,13,15,17,23,25 (ugly),33
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Nonhomogeneous: Undetermined Coeff.
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4.5
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3,7,17,19,23,25,27,33 (trig ident for cos^3),35
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Nonhomogeneous: General solutions
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4.6
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1,3,5,7,11,13,17(longish)
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Variation of Parameters
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4.7
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9,11,13,15,17,19, Reduction of Order: 45,47
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Cauchy-Euler equations, Reduction of Order
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4.8
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not covered
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Qualitative theory
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4.9
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1,7,9,11
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Mechanical Vibrations
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4.10
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Not on exam
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Mechanical Vibrations: Forced
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Midterm 2
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Chapter 4 on HW material assigned
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5
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Time permitting at end of course
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Phase Plane, Numerical
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6
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not covered
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General Theory of Linear Equations
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May be used on quizzes and final exam
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7.2
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3,5,9,11,13,15,17
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Laplace Transform Definition
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7.3
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1,3,5,7,9,13,25,31
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Laplace Transform Properties
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7.4
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1,3,7,9,21,23,25 (last 3 are nastier),33,35
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Laplace Transform Inverse
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7.5
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1,3,7(nasty),11 (set y(t)=w(t-2)),15,17,19,35
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Laplace Transform Initial Value Problems
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7.6
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not covered
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Laplace Transform Discontinuous Functions
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7.7
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1,2,3,5,7,9,13
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Laplace Transform Convolution Theorem
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7.8
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not covered
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Laplace Transform - delta function
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7.9
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not covered
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Laplace Transform - Systems of Equations
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8
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not covered
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Series Approximations and Solutions
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9.1
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1,3,5,8,11
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Differential Equations as Systems
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9.2
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none
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Gaussian Elimination
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9.3
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1,3,5,7b,7c,8,9,17,21,27,31,33,35,37,39
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Matrix algebra and Calculus
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9.4
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1,3,5,9,13,15,19, 28!!
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Linear Systems - Normal Form
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9.5
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1,3,5,7,11,19,21,31!!
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Linear Systems - Constant Coefficient (Real Case)
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9.6
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1, 3 (given lamba=1),5,13a
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Linear Systems - Constant Coefficient (Complex Case)
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9.7
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11,13,21a
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Linear Systems - Variation of Parameters
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9.8
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Linear Systems - Repeated eigenvalues.
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Final
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Review questions: Laplace Transforms and Systems
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Exam and Quiz Outlines
Quizzes
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Quiz 1
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1.1,1.2,2.2
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Quiz 2
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2.3,2.4,2.6
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(you will be asked to solve an exact,
a linear, a homogenous and a Bernoulli eqn) |
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Quiz 3
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4.2-4.3, 4.4
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Will include general soln of higher order constant coefficient eqns and simple problems
on undetermined coefficients (4.4). Will NOT include theory regarding independence,
Wronskians, etc.
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Quiz 4
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4.6,4.7 and Reduction of Order
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Cauchy-Euler homogeneous, Variation
of parameters, Reduction of Order |
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Quiz 5
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7.2,7.3,7.4,7.5 (not 7.7)
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Bring your Laplace transform table!! You will have to take transforms using tables
and transform properties, inverst transforms and solve IVP. Partial fractions will
be at most "cubic".
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Quiz 6
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Convolutions 7.7,9.4,9.5
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Bring your Laplace transform table for the convolution question. The focus on 9.4-9.5
will
be: can you solve IVPs and can you find general solutions for the distinct eigenvalue case? |
Midterm 1
Sample Problems
The exam will cover material from the following sections of the textbook:
- Section 1.1 ODE definitions and theory
- Section 1.2 IVP explicit/implicit solutions, existence uniqueness
- Section 2.2 Separable Equations
- Section 2.3 Linear Equations
- Section 2.4 Exact Equations
- Section 2.6 Homogeneous Equations and Bernoulli Equations only
- Section 3.2 Mixing Problems (no population problems)
- Section 3.4 Newtonian Mechanics - falling bodies, friction, rockets
Notes:
- You will have to solve a separable, linear, exact, homogeneous and Bernoulli equation. This forms the bulk of the exam ( about 70%)
- There will be one application problem, either a mixing problem or a rocket problem (15%)
- One question will require you to categorize types of differential equations (15%).
- The sample Problems are a good indication of the difficulty level of the problems.
Midterm 2
Sample Problems
The exam will cover material from the following sections of the textbook: 4.2-4.7, 4.9
- Constant Coefficient 2nd order homogeneous yh(t)
- Constant Coefficient 3rd order homogeneous yh(t) with one solution known
- Constant Coefficient 2nd order: Undetermined Coefficients Method for yp(t)
- General Solutions y(t)=yh(t) + yp(t), Initial Value Problems, Wronskian for independence
- Cauchy Euler 2nd Order homogeneous yh(t)
- Variation of Parameter Method for yp(t) - standard form.
- Reduction of order: homogeneous solution y1(t) from given homogeneous y1(t)
- Mechanical Vibrations: Amplitude Phase Form y= A sin(wt+phi) for unforced case
Notes:
- There will be an amplitude-phase problem (10-15%). In fact, there will be a question from each point 1-8 above with the sole possible exception of 2.
- The sample problems are a good indication of the difficulty level of the problems but this sheet has only one amplitude-phase problem.
- Undetermined coefficients is ONLY for L(y)=ay''+by'+cy=f and not L(y)=ax2y''+bxy'+cy=f
Final
Important: Thursday, Dec 10, 10:00-11:50am, ROBH 218
- Sections of the textbook covered: 7.2-7.5, 7.7, 9.4-9.7 and repeated eigenvalues notes
- Sample Problems: Laplace Transforms (Ch 7) and Systems (Ch 9)
- Laplace Transform Table is allowed on the exam
- The exam will be about 50% on systems and about 50% on Laplace transforms
Topics Covered
- Laplace: using definition to calculate F(s) for discontinuous functions
- Laplace: Taking transforms using tables and properties
- Laplace: Inversion via partial fractions and completing the square
- Laplace: Solving Initial Value Problems
- Laplace: Using convolution theorem to solve IVP and invert transforms
- Systems: Matrix inverse (2x2) and basic matrix calculus: (AX)'=AX'+A'X
- Systems: Independence, Wronskian, Fundamental Matrix X(t)
- Systems: General Solution for homogeneous/nonhomogeneous systems
- Systems: Solving Initial Value Problems using fundamental matrix X(t)
- Systems: Constant A (2x2): real distinct eigenvalues
- Systems: Constant A (2x2): real repeated eigenvalues
- Systems: Constant A (2x2): complex eigenvalue
- Systems: Variation of Parameters
Updated on: 08/07/2015.