MATH 242 Fall 2018. Section 001
Meeting Times
Section  Schedule  Location  

001  MWF  09:40 AM  10:30 AM  LeConte 112 
Instructor information
Textbook
Important deadlines you need to know
General Dates  

Classes begin  Aug 23, 2018 
Labor Day Holiday  Sep 3, 2018 
Fall Break  Oct 18—19, 2018 
General Election Day  Nov 6, 2018 
Thanksgiving Recess  Nov 21—25, 2018 
Last Day of Classes  Dec 7, 2018 
Academic Deadlines  
Last Day to Change/Drop without W  Aug 29, 2018 
First Day W Grade Assigned  Aug 30, 2018 
Last Day to Drop/Withdraw without WF  Oct 15, 2018 
First Day WF Grade Assigned  Oct 16, 2018 
Prerequisites
A grade of C or better in MATH 142
Course Structure and Grading Policies
Your final grade will be computed as follows:

Inclass tests:
There will be four inclass tests scheduled as follows:Test # Date 1 Mon Oct 01 2 Wed Oct 17 3 Fri Nov 02 4 Mon Nov 19 
Final Exam:
If you have taken at least three of the inclass exams, and are unhappy with your potential final score (as computed with the formula above), notify me by email on Monday, November 26, before 6:00 PM. You will have an opportunity to change your course grade by taking a (comprehensive) final exam. The score of the final will substitute your previous grade.
The final exam is scheduled on Friday, December 14, at 9:00 AM.
 Participation in an authorized University activity (such as musical performances, academic competitions, or varsity athletic events in which the student plays a formal role in a University sanctioned event)
 Required participation in military duties
 Mandatory admission interviews for professional or graduate school which cannot be rescheduled
 Participation in legal proceedings or administrative duties that require a student’s presence
 Death or major illness in a student’s immediate family
 Illness of a dependent family member
 Religious holy day if listed on www.interfaithcalendar.org
 Illness that is too severe or contagious for the student to attend class
 Weatherrelated emergencies
The course grade will be determined as follows:
GRADE  RANGE 

A  90%100% 
B+  86%89% 
B  80%85% 
C+  76%79% 
C  70%75% 
D+  66%69% 
D  60%65% 
F  below 60% 
Further Information
 Honor Code: The Honor Code applies to all work for this course. Please review the Honor Code at [this link]. Students found violating the Honor Code will be subject to discipline.
 Class notes and other additional material will be stored in Dropbox. In that case, you may need an account to retrieve it. If you do not have one already, signin through [this link] with your academic email address to receive a base 4GB storage, plus an extra 500MB, free of charge.
 Remember to change your email address on Blackboard if necessary [blackboard.sc.edu]
 Student Disability Resource Center: If you have special needs as addressed by the Americans with Disabilities Act and need any assistance, please notify the instructor immediately.
 Student Success Center:
In partnership with University of South Carolina faculty, the Student Success Center (SSC) offers a number of programs to assist you in better understanding your course material and to aid you on your path to success. SSC programs are facilitated by trained undergraduate peer leaders who have previously excelled in their courses. Resources available to students in this course include: Peer Tutoring: You can make a oneonone appointment with a peer tutor by going to www.sc.edu/success. Dropin Tutoring and Online Tutoring may also be available for this course. Visit the previous website for a full schedule of times, locations, and courses.
 Success Connect: I may communicate with the SSC regarding your progress in the course. If contacted by the SSC, please schedule an appointment to discuss campus resources that are available to you. Success Connect referrals are not punitive and any information shared by me is confidential and subject to FERPA regulations.
Learning Outcomes
Many of the principles or laws underlying the behavior of the natural World are statements or relations involving rates at which things happen. When expressed in mathematical terms, the relations are equations and the rates are derivatives. Equations containing derivatives are called differential equations. Therefore, to understand and to investigate different problems it is necessary to be able to solve or study differential equations.
Some examples of situations where this happens involve the motion of particles, the flow of current in electric circuits, the dissipation of heat in solid objects, the propagation and detection of seismic waves, or the change of populations.
We will focus mainly in the resolution of some particular kind of differential equations. In the case where we are not able to solve them, we will learn numerical approaches to obtain approximations to the solutions.
Summarizing: A student who successfully completes Elemental Differential Equations (MATH 242) will be able to master concepts and gain skills needed to accomplish the following:
 Solve initial value problems and find general or particular solutions to ordinary differential equations of the following types:
 Separable
 Exact
 Nonlinear homogeneous
 First and higherorder linear equations, both homogeneous and inhomogeneous, especially those with constant coefficients
 Develop skill at using solution methods such as
 integrating factors
 substitution
 variation of parameters
 undetermined coefficients
 Laplace transform
 approximations
 Use differential equations to solve problems related to
 population models
 exponential growth
 logistic growth
 harvesting
 Torricelli's Law
 acceleration/velocity
 mixture
 cooling
 mechanical vibrations
 electrical circuits.
 population models
Lesson Plan
Introduction to Differential Equations
Classical Methods

Fri Aug 24
General Introduction to Differential Equations
[p.8 #126]

Mon Aug 27 [Review: Integration]
Integrals as general and particular solutions
[p.15 #110]

Wed Aug 29
—Slope fields and numerical approximation
—Euler's method
[p.25 #110; p.113 #1,4,6,10]

Fri Aug 31 [slides]
Separable equations
Singular Solutions
[p.40 #128]

Wed Sep 05 [slides]
Homogeneous equations
[p.69 #2,3,710,1214]

Fri Sep 07 [slides]
Linear firstorder differential equations
Bernoulli equation
[p.53 #121 and the equations below]
 \( xy' +y = y^2\ln x \)
 \( y'+y \dfrac{x+\tfrac{1}{2}}{x^2+x+1} = \dfrac{(1x^2)y^2}{(x^2+x+1)^{3/2}} \)
 \( (1+x^2) y' =xy+x^2y^2 \)
 \( x^2y'+2x^3y=y^2(1+2x^2) \)
 \( 3y'+y \dfrac{x^2+a^2}{x(x^2a^2)} = \dfrac{1}{y^2} \dfrac{x(3x^2a^2)}{x^2a^2} \)
 \( y' + \dfrac{y}{x+1} = \frac{1}{2} (x+1)^3 y^2 \)

Mon Sep 10 [slides]
(Today's lesson is offered online through [webconnect]. Please log in as a guest)
General substitution methods
[p.69 #1,46,1518]

Wed Sep 12 [slides]
(Today's lesson is offered online through [webconnect]. Please log in as a guest)
Exact equations
[p.69 #3142]

Fri Sep 14 [slides]
Reducible Secondorder Differential Equations
[p.69 #4354]

Mon Sep 17 [slides]
Intro to secondorder linear differential equations
[p.147 #116]

Wed Sep 19 [slides]
Homogeneous linear secondorder differential equations with constant coefficients
[p.170 #3342]

Fri Sep 21 [slides]
Particular solutions for Secondorder linear differential equations with constant coefficients
—The method of variation of parameters.
[p.195 #120, 4756] Use exclusively the method of variation of parameters

Mon Sep 24 [slides]
Particular solutions for Secondorder linear differential equations with constant coefficients
—The method of undetermined coefficients (Part I: the easy examples)
[No HW today]

Wed Sep 26 [slides]
Particular solutions for Secondorder linear differential equations with constant coefficients
—The method of undetermined coefficients (Part II: the hard examples).
[At this point, you should be able to do problems #156 in p.210 using either method]

Fri Sep 28
General solutions to Secondorder linear differential equations with constant coefficients
Review

Mon Oct 01
First Test. Classical Methods
Methods based on the Laplace Transform

Wed Oct 03 [slides]
Improper integrals revisited
[p.450 #1132]
Find the Laplace transform of \( f(x) = \cos \beta x \), and \( f(x) = 1/\sqrt{x} \) using the definition. 
Fri Oct 05 [slides]
The Gamma function
[p.462 #116]

Mon Oct 08 [slides]
—Linearization of Transforms
—Differentiation of Transforms
—Translation on the saxis
[p.462 #1722; p.472 #122, 2738]
Use the table of transforms to find the Inverse Laplace Transform of the following functions: \( F(s) = \dfrac{3}{s^4}, (s>0) \)
 \( F(s) = \dfrac{5}{s+5}, (s>5) \)
 \( F(s) = \dfrac{3}{s4}, (s>4) \)
 \( F(s) = \dfrac{3s+1}{s^2+4}, (s>0) \)
 \( F(s) = \dfrac{53s}{s^2+9}, (s>0) \)
 \( F(s) = \dfrac{9+s}{4s^2}, (s>2) \)
 \( F(s) = \dfrac{1}{s(s3)}, (s>3) \)
 \( F(s) = \dfrac{3}{s(s+5)}, (s>0) \)
 \( F(s) = \dfrac{1}{s(s^2+4)}, (s>0) \)
 \( F(s) = \dfrac{2s+1}{s(s^2+9)} , (s>0) \)
 \( F(s) = \dfrac{1}{s(s^29)}, (s>3) \)
 \( F(s) = \dfrac{1}{s(s+1)(s+2)}, (s>0) \)
 \( F(s) = \dfrac{2(s4)+3}{(s4)^2+25}), (s>4) \)
 \( F(s) = \dfrac{5s6}{s^23s}, (s>3) \)
 \( F(s) = \dfrac{5s4}{s^3s^22s}, (s>2) \)
 \( F(s) = \dfrac{1}{s^416}, (s>2) \)
 \( f(x) = x^4 e^{\pi x} \)
 \( f(x) = e^{2x} \sin (3\pi x) \)

Wed Oct 10 [slides]
—Integration of Transforms
—The Convolution property
[p.481 #116]
Find the Laplace transform of \( f(x) = \sin 3x \cos 3x \). Hint: Use the formula for the trigonometric function of the double angle.
Find the inverse Laplace transform of \( F(s) = (s^2 + 4)^{2} \). Hint: \( F \) looks like the derivative of another function. 
Fri Oct 12 [slides]
—Laplace transform of derivatives
—Transformation of Initial Value Problems

Mon Oct 15
Review

Wed Oct 17
Second Test. Methods based on Laplace Transform
Applications to Mathematical Modeling

Mon Oct 22
Geometric Applications

Wed Oct 24
More Geometric Applications

Fri Oct 26
Population models
—Introduction to population models
[p.87 #912, 2124]

Mon Oct 29
More population models
—Equilibrium solutions and stability
[p.98 #118 For all these problems, solve the equation explicitly (finding the equilibria), compute a few particular solutions around the equilibria using Maple/Mathematica, and state the stability from this information]

Wed Oct 31
Review

Fri Nov 02
Third Test. Geometric Applications and Population Models

Mon Nov 05
Accelerationvelocity models (Part I)
[p.18 #2429,33,37]

Wed Nov 07
Accelerationvelocity models (Part II)
[p.108 #710,1720]

Fri Nov 09
Mechanical vibrations
—Free undamped motion
[p.195 #14 and if you are brave, try 10,11]

Mon Nov 12
Mechanical vibrations
—Free damped motion
[p.195 #1323]

Wed Nov 14
Mechanical vibrations
—Undamped forced oscillations
[p.222 #16]

Fri Nov 16
Mechanical vibrations
—Damped forced oscillations
Electrical circuits
[at this point, you should be able to solve all problems in page 222. p.231 #110]

Mon Nov 19
Fourth Test. Acceleration/Velocity and Mechanical Vibrations

Mon Nov 26
Applications of Torricelli's Law
[p.44 #5465]
Reviews

Wed Nov 28 [Review (15)]
Integration skills

Fri Nov 30 [Review (25)]
Classical Methods

Mon Dec 03 [Review (35)]
Methods based on Laplace Transform

Wed Dec 05 [Review (45)]
Geometric Applications, Population Models

Fri Dec 07 [Review (55)]
Acceleration/Velocity, Mechanical Vibrations, Torricelli