# Section 001

## Instructor

Francisco Blanco-Silva

**e-mail:** `blanco at math dot sc dot edu`

**office:** LeConte 307

## Meeting Times

Lectures: |
MWF 3:30 AM - 4:20 PM |
LeConte 113 |

Office Hours: |
TTh 3:00 PM — 6:00 PM |
LeConte 307 |

## Important deadlines you need to know

The semester begins Thursday, August 22^{nd}, and ends Friday, December 6^{th}.

The deadline to drop/add and the last day to change credit/audit is Wednesday, August 28^{th}. The first day in which a "W" grade is assigned is therefore Thursday, August 29^{th}.

The last day to obtain a "W" grade or to elect a pass/fail grade is Friday, October 11^{th}. The first day in which a "WF" grade is assigned is therefore Saturday, October 12^{th}.

## Prerequisites

Qualifications through Placement or a grade of **C** or better in MATH 142

## Text

*Differential Equations: Computing and Modeling* by C. Henry Edwards and David E. Penney. **Prentice Hall** 2008 (fourth edition)

Differential Equations Computing and Modeling (4th Edition) |
Student Solutions Manual |

## Course Structure and Grading Policies

Homework problems will be assigned at the end of each lecture; however, they will not be collected and graded. They serve as a guideline to understand the type of problems that will appear on your exams. Your final score for the course will be computed as follows:

- Midterms: each test amounts to 15% of the final grade, for a total of 60% of the course grade. There will be four in-class midterm exams scheduled as follows:

Test # Date **1**Fri Sep 20 **2**Fri Oct 04 **3**Fri Oct 25 **4**Wed Nov 06 (25%) Wed Nov 13 (25%) Wed Nov 20 (25%) Mon Dec 02 (25%) Notice that the fourth midterm is broken in four parts; each of them carrying 25% of the total grade of this midterm.

**You can only take each part of the test if you submit the corresponding HW the day of the exam**. Incomplete or late HW will result in the loss of the corresponding part of the exam, since you will not be allowed to take it.No make-up tests will be given. Only medical, death in the family, religious or official USC business reasons are valid excuses for missing a test and must be verified by letter from a doctor, guardian or supervisor.

- Final exam: 40% of the course grade.The final exam is scheduled on Friday, December 13
^{th}, from 12:30 PM to 3:00 PM.

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.- Some material will be stored in Dropbox. In that case, you will need an account to retrieve it. If you do not have one already, sign-in through [this link] with your academic e-mail address to receive a base 4GB storage, plus an extra 500MB, free of charge.
- Remember to change your e-mail address on Blackboard if necessary [blackboard.sc.edu]
**ADA**: If you have special needs as addressed by the*Americans with Dissabilities Act*and need any assistance, please notify the instructor immediately.**Peer Tutoring:**Tutoring is available for this course to assist you in better understanding the course material. The Peer Tutoring Program at the Student Success Center provides free peer-facilitated study sessions led by qualified and trained undergraduate tutors who have previously taken and excelled in this course. Sessions are open to all students who want to improve their understanding of the material, as well as their grades. Tutoring is offered Sunday 6-10pm and Monday through Thursday 2-9pm. All tutoring sessions will take place on the Mezzanine Level of the Thomas Copper Library unless otherwise noted. Please visit www.sc.edu/tutoring to find the complete tutoring schedule and make an appointment. You may also contact the Student Success Center at 803-777-1000 and tutoring@sc.edu with additional questions. The tutor for your course is Alexandra Ruppe

## 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 higher-order linear equations, both homogeneous and inhomogeneous, especially those with constant coefficients
- Systems of two differential equations

- 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, harvesting, competing species, prey-predator situations, etc), Torricelli's Law, acceleration/velocity, mixture, cooling, mechanical vibrations, or electrical circuits.

## Lesson Plan, HW Assignments, Exams and Project Deadlines

**First Part: Introduction to Differential Equations****Fri Aug 23**: 1.1. General Introduction to Differential Equations [p.8 #1--26]**Mon Aug 26**: [Review: Integration] 1.1 & 1.2. Integrals as general and particular solutions. [p.9 #27--36; p.17 #1--10]**Wed Aug 28**: 1.3 & 2.4. Slope fields and numerical approximation. Euler's method [p.27 #1--10; p.121 #1,4,6,10]**Fri Aug 30**: 1.5. Improved Euler's Method [p.132 #1--10,27,28]**Wed Sep 04**: [slides] 1.4. Separable equations. Singular Solutions. [p.43 #1--28]**Fri Sep 06**: [slides] 1.6. Homogeneous equations. [p.74 #2,3,7--10,12--14]**Mon Sep 09**: [slides] 1.5 & 1.6. Linear first-order differential equations. Bernoulli equation [p.56 #1--21 and the equations below]$latex \begin{array}{ll}

(1)\quad xy' +y = y^2\ln x & (4)\quad x^2y'+2x^3y=y^2(1+2x^2) \\ \\

(2)\quad y'+y\displaystyle{\frac{x+\tfrac{1}{2}}{x^2+x+1}}= \displaystyle{\frac{(1-x^2)y^2}{(x^2+x+1)^{3/2}}} & (5)\quad 3y'+y\displaystyle{\frac{x^2+a^2}{x(x^2-a^2)}}=\displaystyle{\frac{1}{y^2} \frac{x(3x^2-a^2)}{x^2-a^2}} \\ \\

(3)\quad (1+x^2) y' =xy+x^2y^2 & (6)\quad y' + \displaystyle{\frac{y}{x+1}}=-\frac{1}{2} (x+1)^3 y^2\end{array}$**Wed Sep 11**: [slides] 1.6. General substitution methods. [p.74 #1,4--6,15--18]**Fri Sep 13**: [slides] 1.6. Exact equations [p.74 #31--42]**Mon Sep 16**: [slides] 1.6. Reducible Second-order Differential Equations [p.74 #43--54]**Wed Sep 18**: [slides] 3.1. Intro to second-order linear differential equations [p.158 #1--16]**Fri Sep 20**: First Midterm. Chapter 1**Mon Sep 23**: [slides] Homogeneous linear second-order differential equations with constant coefficients [p.158 #33--42]]**Wed Sep 25**: [slides] 3.5. Particular solutions for Second-order linear differential equations with constant coefficients: the method of variation of parameters. [p.210 #1--56] Use exclusively the method of variation of parameters**Fri Sep 27**: [slides] 3.5. Particular solutions for Second-order linear differential equations with constant coefficients: the method of undetermined coefficients (Part I: the easy examples) [No HW today]**Mon Sep 30**: [slides] 3.5. Particular solutions for Second-order linear differential equations with constant coefficients: the method of undetermined coefficients (Part II: the hard examples). General solutions to Second-order linear differential equations with constant coefficients [At this point, you should be able to do problems #1--56 in p.210 using both methods]**Wed Oct 02**: [slides] 7.1. Laplace transform: Improper integrals revisited. [p.450 #11--32. Find the Laplace transform of $latex \cos ax$, and $latex \sqrt{x}$ using the definition]**Fri Oct 04**: Second Midterm. Chapter 3**Mon Oct 07**: [slides] 7.2. Laplace transform: The Gamma function. [p.462 #1--16]**Wed Oct 09**: [slides] 7.4. Laplace transform: Linearization, Differentiation of Transforms and Translation on the s-axis. [p.462 #17--22; p.472 #1--22, 27--38;]**Fri Oct 11**: [slides] 7.3. Laplace transform: Integration of Transforms. The Convolution property [ p.481 #1--16]**Mon Oct 14**: [slides] 7.2. Laplace transform: Laplace transform of derivatives. Transformation of Initial Value Problems**Wed Oct 16**: [slides] 4.1. Systems of differential equations: Introduction. Reduction to first-order systems. [p.255 #1--20, but do not produce the direction fields nor typical solution curves yet]**Mon Oct 21**: [slides] 4.2. Systems of differential equations: Numerical Methods. [p.255 Using what you did in the previous assignment, produce the direction fields and typical solution curves for the systems #10--20; p.266 #1--19]**Wed Oct 23**: Review for third midterm.**Fri Oct 25**: Third Midterm. Chapters 7 and 4**Mon Oct 28**: ---**Wed Oct 30**: Remake Third Midterm

**Second Part: Applications to Mathematical Modeling****Fri Nov 01**: 1.2 & 2.3. Acceleration-velocity models (Part I) [p.18 #24--29,33,37]**Mon Nov 04**: 2.3. Acceleration-velocity models (Part II) [p.108 #7--10,17--20]**Wed Nov 06**: Fourth Midterm (1/3) Acceleration/Velocity**Fri Nov 08**: 2.1. Population models [p.87 #9--12, 21--24]**Mon Nov 11**: 2.2. More population models. Equilibrium solutions and stability [p.98 #1--18 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 Nov 13**: Fourth Midterm (2/3) Population Models**Fri Nov 15**: 3.4. Mechanical vibrations: Free undamped motion [p.195 #1--4 and if you are brave, try 10,11]**Mon Nov 18**: 3.4. Mechanical vibrations: Free damped motion [p.195 #13--23]**Wed Nov 20**: Fourth Midterm (3/3) Mechanical Vibrations I**Fri Nov 22**: Class canceled**Mon Nov 25**: 3.6. Mechanical vibrations: Undamped forced oscillations [p.222 #1--6]**Mon Dec 02**: 3.6. Mechanical vibrations: Damped forced oscillations. 3.7. Electrical circuits [at this point, you should be able to solve all problems in page 222. p.231 #1--10]**Wed Dec 04**: 1.4. Applications of Torricelli's Law [p.44 #54--65]**Fri Dec 06**: Review

**Wed Dec 13**: 12.30PM--3:00PM Final Exam. Chapters 1, 2, 3, 4 and 7.