MATH 242 Spring 2018
Sections 005 and 010
Section  Schedule  Location  

005  TTh  11:40 AM  12:55 PM  LeConte 405 
010  TTh  2:50 PM  4:05 PM  LeConte 112 
Important deadlines you need to know
General Dates  

Classes begins  January 16, 2018 
Last day of classes  April 30, 2018 
Academic Deadlines  
Last Day to Change/Drop  January 22, 2018 
First Day 'W' Grade Assigned  January 23, 2018 
Last Day 'W' Grade Assigned  March 9, 2018 
First Day 'W/F' Grade Assigned  March 10, 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 Tue Feb 20 2 Thu Mar 08 3 Thu Mar 29 4 Thu Apr 12 
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 Thursday, April 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. Section 005: Thursday, May 3, 2018. 12:30 PM
 Section 010: Tuesday, May 8, 2018. 12:30 PM
 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+  85%89% 
B  80%84% 
C+  75%79% 
C  70%74% 
D+  65%69% 
D  60%64% 
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

First part: Introduction to Differential Equations

Tue Jan 16: 1.1, 1.2. [Review: Integration]
General Introduction to Differential Equations.
Integrals as general and particular solutions.
[p.8 #131,3742, p.15 #110] 
Thu Jan 18: 1.3, 2.4, 2.5.
Slope fields.
Numerical Approximation: Euler's methods.
[p.24 #110, p.113 #1,4,6,10] 
Tue Jan 23: 1.4, 1.6.
[slide  slide]
Separable equations. Singular Solutions.
Homogeneous FirstOrder equations.
[p.40 #128, p.69 #2,3,710,1214] 
Thu Jan 25: 1.5, 1.6.
[slides]
Linear firstorder differential equations. Bernoulli equation.
[p.53 #125, and the equations below] \( xy' +y = y^2\ln x \)
 \( y'+y \dfrac{x+\tfrac{1}{2}}{x^2+x+1} = \dfrac{(1x^2)y^2}{(x^2+x+1)^{3/2}} \)
 \( (1+x^2) y' =xy+x^2y^2 \)
 \( x^2y'+2x^3y=y^2(1+2x^2) \)
 \( 3y'+y \dfrac{x^2+a^2}{x(x^2a^2)} = \dfrac{1}{y^2} \dfrac{x(3x^2a^2)}{x^2a^2} \)
 \( y' + \dfrac{y}{x+1} = \frac{1}{2} (x+1)^3 y^2 \)

Tue Jan 30: 1.6.
[slides]
General Substitution Methods.
[p.74 #1,46,1518] 
Thu Feb 01: 1.6
[slideslides]
Exact equations. Reducible Secondorder Differential Equations.
[p.74 #3154] 
Thu Feb 06: 3.1, 3.2, 3.3.
[slides]
Intro to linear differential equations of Higher Order.
SecondOrder Linear Equations.
[p.147 #116] 
Tue Feb 08: 3.5.
[slide  slide  slide]
Homogeneous linear secondorder differential equations with constant coefficients.
Particular solutions for Secondorder linear differential equations with constant coefficients:
 Variation of parameters.
[p.147 #3342, p.170 #19,21,22,23, p.195 #1,2,3,6,10] 
Thu Feb 13: 3.5.
[slide]
Particular solutions for Secondorder linear differential equations with constant coefficients:
 Undetermined coefficients.
[p.195 #21,23,26,3135,5355]  Thu Feb 15: Review
 Tue Feb 20: First Test

Tue Jan 16: 1.1, 1.2. [Review: Integration]

Second Part: Laplace Transform

Thu Feb 22: 7.1, 7.2.
[slides]
Improper integrals revisited. The Laplace Transform
[p.445 #2,3,4,6,35]
Find the Laplace transform of \( f(x) = \cos \beta x \), and \( f(x) = 1/\sqrt{x} \) using the definition. 
Tue Feb 27: 7.4, 7.3
[slide  slide]
The Gamma Function.
Linearization and Translation on the saxis.
Assignment
Use the table of transforms to find the Inverse Laplace Transform of the following functions: \( F(s) = \dfrac{3}{s^4}, (s>0) \)
 \( F(s) = \dfrac{5}{s+5}, (s>5) \)
 \( F(s) = \dfrac{3}{s4}, (s>4) \)
 \( F(s) = \dfrac{3s+1}{s^2+4}, (s>0) \)
 \( F(s) = \dfrac{53s}{s^2+9}, (s>0) \)
 \( F(s) = \dfrac{9+s}{4s^2}, (s>2) \)
 \( F(s) = \dfrac{1}{s(s3)}, (s>3) \)
 \( F(s) = \dfrac{3}{s(s+5)}, (s>0) \)
 \( F(s) = \dfrac{1}{s(s^2+4)}, (s>0) \)
 \( F(s) = \dfrac{2s+1}{s(s^2+9)} , (s>0) \)
 \( F(s) = \dfrac{1}{s(s^29)}, (s>3) \)
 \( F(s) = \dfrac{1}{s(s+1)(s+2)}, (s>0) \)
 \( F(s) = \dfrac{2(s4)+3}{(s4)^2+25}), (s>4) \)
 \( F(s) = \dfrac{5s6}{s^23s}, (s>3) \)
 \( F(s) = \dfrac{5s4}{s^3s^22s}, (s>2) \)
 \( F(s) = \dfrac{1}{s^416}, (s>2) \)
 \( f(x) = x^4 e^{\pi x} \)
 \( f(x) = e^{2x} \sin (3\pi x) \)

Thu Mar 01: 7.2.
[slide  slide]
Differentiation of Transforms.
Laplace transform of derivatives.
Transformation of Initial Value Problems.
[p.456 #110, p.464 2730, p.473 #15,16,17] 
Tue Mar 06: 7.3
[slides]
Integration of Transforms.
The Convolution property
[ p.481 #116]
Review
[p.446 #11,12,14, p.464 #2,4]
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.  Thu Mar 08: Second Test

Thu Feb 22: 7.1, 7.2.
[slides]

Third Part: Applications to Mathematical Modeling

Tue Mar 20: 2.1, 2.2.
Geometric Applications
[Notes and Homework] 
Thu Mar 22: 1.2, 2.3.
Population models
Equilibrium solutions and stability
[p.40 #33,34,37, p.82 #15,16,21,22,23,26]  Tue Mar 27: Integration Challenge
 Thu Mar 29: Review
 Tue Apr 03: Third Test

Thu Apr 05: 3.4.
Mechanical vibrations:
 Free undamped motion
 Free damped motion
[p.181 #14,13] 
Tue Apr 10: 3.6.
Mechanical vibrations:
 Undamped forced oscillations
 Damped forced oscillations
[p.181 #1521, p.206 #14]  Thu Apr 12: Review
 Tue Apr 17: Fourth Test

Thu Apr 19: 1.4.
Applications of Torricelli's Law
[p.42 #5464]

Tue Mar 20: 2.1, 2.2.

Reviews
 Tue Apr 24: Review (12)
 Tue Apr 26: Review (22)