Classes

Instructor

Jacob Rosen
Office: Engineering IV Building, Room 37-146
Voice Office: 831.459.5302
e-mail: jacobrosen@ucla.edu
Anonymous Email (For the Subject Line use: MAE 82A)
Office Hours: Wed 2:00-3:00

MAE 82

Mathematics of Engineering

Course Summary: Methods of solving ordinary differential equations in engineering. Review of matrix algebra. Solutions of systems of first- and second-order ordinary differential equations. Introduction to Laplace transforms and their application to ordinary differential equations. Introduction to boundary value problems. Nonlinear differential equations and stability

Prerequisite: (Matlab)

Table of Contents

Assignments & Grading
Textbooks
TA Office Hours
Syllabus
Book Notes / Problems (11th Edition)
Homework Problems / Assigned Reading
Class Notes
Video Clips (YouTube)
Supplement References
Matlab / Simulink

Assignments & Grading

Plan Load	Content	Grading
A High	Textbook Pro. - Part 1 & 2 Project Matlab Required: Yes	Problem Sets - 20% Projects - 9% Attendance - 5% Exam 1 Mid - 33% Exam 2 Final - 33%
B Med	Textbook Pro. - Part 1 Project Matlab Required: Yes	Problem Sets - 10% Projects - 9% Attendance - 5% Exam 1 Mid - 38% Exam 2 Final - 38%
C Med	Textbook Pro. - Part 1 & 2 Matlab Required: No	Problem Sets - 20% Attendance - 5% Exam 1 Mid - 37.5% Exam 2 Final - 37.5%
D Low	Textbook Pro. - Part 1 Matlab Required: No	Problem Sets - 10% Attendance - 5% Exam 1 Mid - 42.5% Exam 2 Final - 42.5%

Textbooks

Textbook (Required): Elementary Ordinary Differential Equations and Boundary Value Problems”. William E. Boyce & Richard C. DiPrima, 11th Edition, Wiley 2017
ISBN: 978-1-119-32063-0

Textbook (supplement): Differential Equations with Boundary-Value Problems" Zill et al. 7th / 8th editions, Cengage Learning
ISBN-13: 978-1111827069 ISBN-10: 1111827060

Textbook (supplement): Differential Equations, A Primer for Scientists and Engineers, Christian Constanda

Textbook (supplement): Engineering Diffrential Equations, Theory & Applications, Bill Goodwine - Errata

Textbook (supplement): Differential Equations and Thier Applications, Martin Braun

TA Office Hours

Changyeob Shin (TA)
Office: Engineering IV
Office Hours: Mon 13:00 - 15:00; ENG IV 43-147
E-mail: shinhujune@ucla.edu

Yi-jui Chang (TA)
Office: Engineering IV
Office Hours: Wed 15:00 - 17:00; ENG IV 43-147
E-mail: michaelchang0928@gmail.com

Syllabus

1. First & Second order Linear Differential equations

1.1 Classifications of differential equations, linear/nonlinear, order

1.2 First order Linear Differential equations
1.2.1 Method of integrating factors
1.2.2 Variation of parameters

1.3. First Order non linear Differential Equations
1.3.1 Separable equations
1.3.2 Integrating factor
1.3.3 Change of variables and parametric solutions

1.4 Second order Linear Differential equations
1.4.1 Homogeneous equations with constant coefficients
1.4.2 Characteristic equation, distinct and repeated solutions
1.4.3 Nonhomogeneous equations, method of undetermined coefficients and variation of parameters
1.4.4 Applications to selected engineering problems

2. High order linear equations

2.1 Characteristic equation. Consider distinct and repeated solutions
2.2 Variation of Parameters (Wronskian)
2.3 Undetermined coefficients

[Mid Term Exam]

3. Series solution method

3.1 Power series solution for a linear equation (general n-th order) with constant coefficients
3.2 Introduction to the use of power series when coefficients are not constant
3.3 Series solution near an ordinary point
3.4 Series solution near a regular point
3.5 Error and accuracy of power series solution representation

4. Laplace Transform

4.1 Definition and convergence of Laplace Transform
4.2 Solution to initial value problems
4.3 Linear differential equations with discontinuous forcing term (Dirac delta, step fiunction)

5. Systems of linear equations

5.1 Review of matrices, eigenvalues, eigenfunctions
5.2 Homogeneous and no homogeneous linear systems with constant coefficients
5.3 Discussion on the function e^At
5.4 Systems of higher order Linear Equations and state space representation
5.5 Nonlinear Systems of Differential equations (Equilibrium points)
5.6 Concepts of stability and steady state Solutions
5.7 Special functions, Dirac Delta, Step function

[Final Exam]

6. Numerical Methods

Numerical Methods for first order equations
7.1 Introduce Euler Method
7.2 Runge-Kutta Method
7.3 Multistep Methods
7.4 Errors and stability

Book Notes / Problems (11th Edition)

Book Chapter - Notes

Clapter 1 - Intoduction
Clapter 2 - First Order Diff Eq.
Clapter 3 - Second Order Linear Diff. Eq.
Clapter 4 - Higher Order Linear Diff. Eq.
Clapter 5 - Series Solution
Clapter 6 - Laplace Transform
Clapter 7 - Systems of First Order Diff. Eq.
Clapter 8 - Numerical Methods
Clapter 9 - Non Linear Diff. Eq. and Stability

Book Problems (11th Edition)

Clapter 1 - Intoduction
Clapter 2 - First Order Diff Eq.
Clapter 3 - Second Order Linear Diff. Eq.
Clapter 4 - Higher Order Linear Diff. Eq.
Clapter 5 - Series Solution
Clapter 6 - Laplace Transform
Clapter 7 - Systems of First Order Diff. Eq.
Clapter 8 - Numerical Methods
Answers to Problems

Homework Problems / Assigned Reading

Week	Reading Assignment & Pre-Class	Class / Homework	Submission
1a	Book Ch. 1	Class Notes Class 00 : Course Info. Class 01 : Introduction. Post Week Homework - HW1 Part 1 P. 8, Pr. 2 P. 8-9, Pr. 11-16 P. 9, Pr. 20 P. 15, Pr. 3 P. 22, Pr. 1 P. 22, Pr. 7 Part 2 P. 8, Pr. 3 P. 9, Pr. 19 P. 15, Pr. 4 P. 22, Pr. 3 P. 22, Pr. 8 Project 1 (Pendulum) HW Due Date - Friday Week 2 (Midnight) Project Due Date - Monday Week 3 (Midnight) () Note 1: Directional Field Example (Matlab) () Note 2: Vectorfiled2D Matlab Help HW 1 Solution Project 1 Solution Project 1 - Matlab (M-File)	Preclass None
1b	Book Ch. 2 Sections 2.1-2.4 Pre - Class Pg. 58 Pr. 26	Class Notes Class 02 : First Order Diff Eq. - Linear	Preclass
S1		Section Notes	None
2a	Book Ch. 2 Sections 2.5 Pre - Class Pg. 67 Pr. 2	Class Notes Class 02 : First Order Diff Eq. - Linear Post Week Homework - HW2 Part 1 P. 31, Pr. 3 P. 38, Pr. 3 P. 39, Pr. 25 P. 48, Pr. 11 P. 68, Pr. 20 P. 75, Pr. 8 Part 2 P. 31, Pr. 12 P. 38, Pr. 6 P. 39, Pr. 25 P. 48, Pr. 14 P. 75, Pr. 21 Project 2 (Alle Effect) () Note: Problem 3 in the project refers to phase portrait which is similar to draw the direction field and integral curves of N vs t. You may refer section 2.5 Autonomous Equations and Population Dynamics on page 58 to 66 of the textbook. The figures 2.5.3, 2.5.6, and 2.5.8 are good examples. HW Due Date - Friday Week 3 (Midnight) Project Due Date - Monday Week 4 (Midnight)* HW 2 Solution Project 2 Solution Project 2 - Matlab (M-File)	Preclass
2b	Book Ch. 2 Sections 2.6, 2.8 Pre - Class Pg. 75 Pr. 1	Class Notes Class 03 : First Order Diff Eq. - Non-Linear.	Preclass Fri - HW1
S1		Section Notes
3a	Book Ch. 3 Sections 3.1-3.4 Pre - Class Pg. 131 Pr. 7	Class Notes Class 04 : Second Order Diff Eq. - Homogeneous MathLets - POLES AND VIBRATIONS Post Week Homework - HW3 Part 1 P. 109, Pr. 9 (real roots) P. 125, Pr. 6 (complex roots) P. 132, Pr. 2 (identical roots) P. 141, Pr. 17 (Non Homo) P. 146, Pr. 11 (Non Homo) P. 168, Pr. 9 (MKC) Part 2 P. 109, Pr. 11 P. 125, Pr. 7 P. 132, Pr. 8 P. 141, Pr. 21 P. 168, Pr. 8 Project 3 (Bungee Jumping) HW Due Date - Friday Week 4 (Midnight) Project Due Date - Monday Week 5 (Midnight) HW 3 Solution Project 3 Solution Project 3 - Matlab (M - File)	Mon - Project 1 Preclass
3b	Book Ch. 3 Sections 3.7 Pre - Class Pg. 158 Pr. 17		Preclass Fri - HW2
4a	Book Ch. 3 Sections 3.5-3.6 Pre - Class Pg. 141 Pr. 1	Post Week Homework - HW4 Part 1 P. 180 Pr. 16 P. 180 Pr. 29 a, b, c P. 184 Pr. 1 P. 188 Pr. 3 Part 2 P. 180 Pr. 9 P. 184 Pr. 2 P. 188 Pr. 9 Project 4 (Tacoma Nerrows) HW Due Date - Friday Week 5 (Midnight) Project Due Date - Monday Week 6 (Midnight) HW 4 Solution Project 4 Solution Project 4 - Matlab (M-File)	Mon - Project 2 Preclass
4b	Book Ch. 3 Sections 3.8 Pre - Class Pg. 167 Pr. 4		Preclass Fri - HW3
5a	Book Ch. 4 Sections 4.1-4.2 Pre - Class Pg. 180 Pr. 2	HW 5 Prep for the midterm using practice exam - No Need to submit P. 31 Pr. 1 P. 75 Pr. 1 P. 167 Pr. 4,5 P. 184 Pr. 12 P. 188 Pr. 3 Project 5 (Beam Buckling) Problem 1 a-e p. 12 Problem 2 a,b p. 15 Problem 3 a p. 16 HW Due Date - No Need to submit Project Due Date - Monday Week 7 (Midnight) Project 5 Solution Project 5 - Matlab (M-File)	Mon - Project 3 Preclass
5b	Book Ch. 4 Sections 4.3-4.4 Pre - Class Pg. 184 Pr. 2		Preclass Fri - HW4
6a			Mon - Project 4 Preclass
6b	Mid Term Thursday	In Class (Open Book & Open Notes)
7a	Book Ch. 5 Sections 5.1 Alternative Read Series Solution (Zill) Series Solution (Xie) Pre - Class Pg. 195 Pr. 7	Post Week Homework - HW6 Part 1 P. 218 Pr. 3 P. 218 Pr. 13 P. 223 Pr. 10 P. 229 Pr. 10 P. 230 Pr. 12 Part 2 P. 218 Pr. 20 P. 223 Pr. 12 P. 229 Pr. 13 Project 6 - None HW Due Date - Friday Week 8 (Midnight) HW 6 Solution	Preclass
7b	Book Ch. 5 Sections 5.2-5.3 Alternative Read Series Solution (Zill) Series Solution (Xie) Pre - Class Pg. 204 Pr. 2		Preclass
8a	Book Ch. 5 Sections 5.5-5.6 Alternative Read Series Solution (Zill) Series Solution (Xie) Pre - Class Pg. 223 Pr. 1		Mon - Project 5 Preclass
8b	Book Ch. 6 Sections 6.1-6.2 Alternative Read Laplace Transform Pre - Class Pg. 255 Pr. 8	Post Week Homework - HW7 Part 1 P. 255 Pr. 6,15,18 P. 263 Pr. 22 P. 268 Pr. 2 P. 269 Pr. 11 P. 273 Pr. 7 P. 274 Pr. 13 P. 279 Pr. 6 Part 2 P. 255 Pr. 19 P. 268 Pr. 6 P. 269 Pr. 12 P. 279 Pr. 9 Project 7 (Bessel) HW Due Date - Friday Week 9 (Midnight) Project Due Date - Monday Week 10 (Midnight) HW 7 Solution Project 7 Solution Project 7 - Matlab (M-File)	Fri-HW6 Preclass
9a	Book Ch. 6 Sections 6.3-6.4 Alternative Read Laplace Transform Pre - Class Pg. 268 Pr. 1		Mon - Project 7 Preclass
9b	Book Ch. 6 Sections 6.5-6.6 Alternative Read Laplace Transform Pre - Class Pg. 273 Pr. 1	Revew at Home - Book Ch. 7.1-7.4 Post Week Homework - HW8 Linear Algebra Review (Review Only No Need To Submit) P. 293 Pr. 4 a b c P. 293 Pr. 5 a b c 6 a b c P. 294 Pr. 8 a b c d P. 294 Pr. 14 P. 297 Pr. 17 P. 303 Pr. 4,7,12,19 Project 8 (Mayfair Diner) HW 8 Solution Project 8 Solution Project 8 - Matlab (M-File)	Preclass Fri- HW7
10a	Book Ch. 7 Sections 7.5-7.8 Alternative Read System of ODE Pre - Class Pg. 343 Pr. 1	Post Week Homework - HW9 Part 1 () P. 318 Pr. 11 (Homo - Regular) P. 328 Pr. 7 (Homo - Complex) P. 344 Pr. 17 (Homo - Repated) P. 351 Pr. 2,3,4 (Non Homo) Note:* Sketck phase portraits for all solutions (See Video) *Part 2 () P. 318 Pr. 12 (Homo - Regular) P. 328 Pr. 8 (Homo - Complex) P. 344 Pr. 18 (Homo - Repated) P. 351 Pr. 1,5,8 (Non Homo) () Note:* Sketck phase portraits for all solutions (See Video) Project 9 (Arms Races) Notes (Typos) 1. Q3a - second eq. - dy/dt 2. Q3b - y=e+fx Due date - Friday Week 10 HW Due Date - Friday Week 10 (Midnight) Project Due Date - Friday Week 10 (Midnight)* HW 9 Solution Project 9 Solution Project 9 - Matlab (M-File)	Mon - HW8 Project 8 Preclass
10b	Book Ch. 7 Sections 7.9 Alternative Read System of ODE Pre - Class Pg. 351 Pr. 1		Preclass Fri or Mon- HW9 Project 9
Exam 2 - Final
		Prep for the Final using practice Exam - No Need to submit P. 218 Pr. 4 P. 218 Pr. 6 P. 223 Pr. 6 P. 229 Pr. 5 P. 268 Pr. 6 P. 273 Pr. 3 P. 273 Pr. 6 P. 351 Pr. 1 P. 351 Pr. 4 P. 352 Pr. 6 P. 352 Pr. 8

Class Notes

Post Class Slides (Annotated)

Class 00 : Course Info.
Class 01 : Introduction.
Class 02 : First Order Diff Eq. - Linear
Class 03 : First Order Diff Eq. - Non-Linear.
Class 04 : Second Order Diff Eq. - Homogeneous
Class 05 : Second Order Diff Eq. - Nonhomogeneous.
Class 06 : Second Order Diff Eq. - Nonhomogeneous - Force Vibration
Class 07 : High Order Diff. Eq. - Homogeneous.
Class 08 : High Order Diff. Eq. - Nonhomogeneous.
Class 09 : Series Soultion - Intro.
Class 10 : Series Soultion - Ordinary Point.
Class 11 : Series Soultion - Regular Singular Point.
Class 12 : Laplace - Intro & Diff Eq.
Class 13 : Laplace - Step Function.
Class 14 : Laplace - Impulse Function.
Class 15 : Laplace - Systems (S- Plane).
Class 15 Appendix : Linear Algebra - Review.
Class 16 : System of First Order Linear Diff. Eq. - Homogeneous.
Class 17 : System of First Order Linear Diff. Eq. - NonHomogeneous.
Class 18 : Numerical Approuch - First Order
Class 19 : Numerical Approuch - System

Pre Class Slides (Blank)

Class 01
Class 02
Class 03
Class 04
Class 05
Class 06
Class 07
Class 08
Class 09
Class 10
Class 11
Class 12
Class 13
Class 14
Class 15
Class 16
Class 17
Class 18
Class 19

TA Sessions

TA1 Session 01 , TA2 Session 01
TA1 Session 02 , TA2 Session 02
TA1 Session 03 , TA2 Session 03
TA1 Session 04 , TA2 Session 04
TA1 Session 05 , TA2 Session 05
TA1 Session 06 , TA2 Session 06
TA1 Session 07 , TA2 Session 07
TA1 Session 08 , TA2 Session 08
TA1 Session 09 , TA2 Session 09
TA1 Session 10 , TA2 Session 10

Video Clips (YouTube)

Class 01 : Introduction
e (Euler's Number) - Numberphile
The number e is everywhere
The number e explained in depth for (smart) dummies
How One Man Changed the High Jump Forever
An athlete uses physics to shatter world records (TED Ed)

Class 03 : First Order Diff Eq. - Linear
Human Population Growth - Crash Course Ecology #3
Overpopulation – The Human Explosion Explained
OVERPOPULATED - BBC Documentary
Human Population Through Time
A Magic Number? - American Scientist

Class 04 - Second Order Diff Eq - Homogeneous
Car Suspention
Hydraulic shock absorbers
Vibration Tests - Pilots
Taipei 101 - Earthquake 2002
Taipei 101 Damper ( Typhoon Soulik)
Boeing 787 Dreamliner wingflex
Wings of the Boeing 787
Two Towers (Model) with a damper
Trailer Sway

Class 05/06 - Second Order Diff Eq - Non Homogeneous
Collapse of the Tacoma Narrows Bridge
Vibrations in a Steel Ruler
Two Towres (Model) earthquake Protection
CSeries Ground Vibration Test (GVT)
CH 650 E Ground Vibration Tests
Chinook CH-47 Helicopter Ground Resonance Test
Braking a Glass
Wine glass resonance in slow motion
SDOF Resonance Vibration Test
Plate Resonance Experiment
Double Pendulum
Square wave
Fourier series

Class 07-08 High Order Diff. Eq. (Beams)
Tensile Testing
Beam - Finite element analysis of arch bridge Bending
Vibrations in Rotor | Resonance | Critical Speed | Whirling
Understanding Resonance Mode Shapes

Class 9-11 Series Solution
Taylor series - Wikipedia

Class 12-15 Laplace / Fourier Transforms
Laplace Transform Explained and Visualized Intuitively
The Laplace Transform - A Graphical Approach
Fourier Transform, Fourier Series, and frequency spectrum
Introduction to the Fourier Transform
Impulse - Golf Ball
Football to the Face
Effects of a Concussion on Brain Cells - Medical Animation
Concussion / Traumatic Brain Injury (TBI)
What happens when an aircraft crashes

Class 16
Eigenvalues and eigenvectors (Wikipedia)
What is an Eigenvector?
Vibration Mode Shapes of a Tall Building
Sketching Phase Portraits

Supplement References

First & Second Order Diff Eq.: ODE 1; ODE 2
Laplace (Tables): Laplace Transform
Laplace Transform
Seriers Solution
System of First Order ODE
The Theory of the Rainbow - Ariy' Diff Eq.

Matlab / Simulink

Matlab Tutorilas [PDF]

MATLAB Tutorial
Simulink
Simulink - Getting Started [PDF]
Simulink - User Guide [PDF]

MATLAB Books
Free access books for UCLA stdents (on campus or via VPN)
MATLAB Mathematical Analysis, César Pérez López
MATLAB Matrix Algebra, César Pérez López
MATLAB Graphical Programming, César Pérez López
MATLAB Differential Equations, César Pérez López

YouTube (Matlab)
The Complete MATLAB Course: Beginner to Advanced

YouTube (Simulink Tutorials)
Introduction to Simulink - Webinar
Simulink: Solving Second Order Differential Equation
Simulink: Solving Second Order Differential Equation (Part 1)
Simulink: Solving Second Order Differential Equation (Part 2)
Simulink: Solving a Set of three First Order Differntial Equations
Simulink: Solving a Set of two Second Order Differntial Equations
Simulink: Dyanmics Ferrari / Fiat (Laplace)

Access to MATLAB (UCLA)

1) Create SEASnet account
www.seasnet.ucla.edu > Accounts > Create Account

(After applying through website, you need to visit SEAS help desk for getting account)

2) Remote Connection
www.seasnet.ucla.edu > LABS > Seasnet Terminal Server > Terminal Server (Remote Desktop Connect) > Follow instructions

Matlab (Scripts) / Simulink (Models)

Intro to Matlab - Matlab Script

Paradoxes & Dilemas

The Prisoner's Dilema
Wikipedia - The Prisoner's dilemma
YouTube - The Prisoner's Dilemma
YouTube - The Iterated Prisoner's Dilemma - Cooperation

The Trolley's Problem
Wikipedia - The Trolley Problem - Wikipedia
YouTube - The Trolley Problem
YouTube - TED-Ed
YouTube - 10 Insane Trolley Problems