# MATH 242. Section 002

## Instructor

Francisco Blanco-Silva

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

**office:** LeConte 307

## Meeting Times

Lectures: |
MWF 1:25 AM - 2:15 PM |
LeConte 112 |

Office Hours: |
TTh 1:00 PM — 4:00 PM |
LeConte 307 |

## Important deadlines you need to know

The semester begins Thursday, August 23^{rd}, and ends Friday, December 7^{th}.

The deadline to drop/add and the last day to change credit/audit is Wednesday, August 29^{th}. The first day in which a "W" grade is assigned is therefore Thursday, August 30^{th}.

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

## 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 21 **2**Fri Oct 05 **3**Fri Oct 26 **4**Mon Nov 12 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 to the instructor.

- Final exam: 40% of the course grade.The final exam is scheduled on Wednesday, December 12
^{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

- 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.**Math Tutoring Center:**The Math Tutoring Center is a free tutoring service for MATH 111, 115, 122, 141, 142, 170, 221, 222, and 241. The center also maintains a list of private tutors for math and statistics. The center is located in LeConte, room 105, and the schedule is available at the Department of Mathematics website [www.math.sc.edu]. No appointment is necessary.**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 24**: 1.1. General Introduction to Differential Equations [p.8 #1--26]**Mon Aug 27**: 1.1 & 1.2. Intro to modeling. Integrals as general and particular solutions. [p.9 #27--36; p.17 #1--10]**Wed Aug 29**: 1.3 & 2.4. Slope fields and numerical approximation. Euler's method [p.27 #1--10; p.121 #1,4,6,10]**Fri Aug 31**: 1.5. Improved Euler's Method [p.132 #1--10,27,28]**Wed Sep 05**: 1.4. Separable equations and applications. [p.43 #1--28,33,37,43]**Fri Sep 07**: 1.6. Homogeneous equations. [p.74 #2,3,7--10,12--14]**Mon Sep 10**: 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 12**: 1.6. General substitution methods. [p.74 #1,4--6,15--18]**Fri Sep 14**: 1.6. Conservative vector fields, and their relationship to differential equations: Exact equations [p.74 #31--42]**Mon Sep 17**: 1.6. Reducible Second-order Differential Equations [p.74 #43--54]**Wed Sep 19**: 3.1. Intro to Higher-order differential equations [p.158 #1--16] Review for first midterm**Fri Sep 21**: First Midterm. Chapter 1 [Practice exam]**Mon Sep 24**: Linear independence of solutions and Wronskians. Homogeneous linear second-order differential equations with constant coefficients [p.158 #20--26,33--42]]**Wed Sep 26**: 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 28**: 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 Oct 01**: 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 03**: 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 05**: Second Midterm. Chapter 3 [Practice exam? Just do all the HW]**Mon Oct 08**: 7.2. Laplace transform: The Gamma function. Laplace transform of derivatives. [p.462 #1--16]**Wed Oct 10**: 7.4. Laplace transform: Differentiation of Transforms [p.462 #17--22; p.481 #15,16]**Fri Oct 12**: 7.3. Laplace transform: Translation of the s-axis. The convolution property [p.472 #1--22, 27--38; p.481 #1--14]**Mon Oct 15**: 7.2. Laplace transform: Integration of Transforms. Putting it all together (transformation of Initial Value Problems)**Wed Oct 17**: 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 22**: 4.2. Systems of differential equations: Slope fields, Euler's method and solution by elimination. [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 24**: Systems of differential equations: Critical points. Functional relationship among variables.**Fri Oct 26**: Third Midterm. Chapters 7 and 4

**Second Part: Applications to Mathematical Modeling****Mon Oct 29**: 1.2 & 2.3. Acceleration-velocity models (Part I) [p.18 #24--29,33,37,39]**Wed Oct 31**: 2.3. Acceleration-velocity models (Part II) [p.108 #7--10,17--20]**Fri Nov 02**: 2.1. Population models [p.87 #9--12, 21--24]**Mon Nov 05**: 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 07**: Competing Species. [Interpret the following systems as describing the interaction of two species with populations $latex x$ and $latex y.$ In each of these problems carry out the following steps: (i) Draw a slope field and describe how solutions seem to behave. (ii) Find the critical points, and determine their stability. (iii) Sketch trajectories in the neighborhood of each critical point.]$latex \begin{array}{rlrl}

(1) & x'=x(1.5-x-0.5y) & (2) & x'=x(1.5-x-0.5y) \\

& y'=y(2-y-0.75x) & & y'=y(2-0.5y-1.5x) \\ \\

(3) & x'=x(1.5-0.5x-y) & (4) & x'=x(1.5-0.5x-y) \\

& y'=y(2-y-1.125x) & & y'=y(0.75-y-0.125x) \\ \\

(5) & x'=x(1-x-y) & (6) & x'=x(1-x+0.5x) \\

& y'=y(1.5-y-x) & & y'=y(2.5-1.5y+0.25x)

\end{array}$**Fri Nov 09**: Predator-Prey models. [Interpret the following systems as describing the interaction of two species with population densities $latex x$ and $latex y.$ In each of these problems carry out the following steps: (i) Draw a slope field and describe how solutions seem to behave. (ii) Find the critical points, and determine their stability. (iii) Sketch trajectories in the neighborhood of each critical point (iv) If possible, find functional relationship between the two variables.]$latex \begin{array}{rlrl}

(1) & x'=x(1.5-0.5y) & (2) & x' = x(1-0.5y) \\

& y'=y(-0.5+x) & & y'=y(-0.25+0.5x) \\ \\

(3) & x'=x(1-0.5x-0.5y) & (4) & x'=x(1.25-x-0.5y) \\

& y'=y(-0.25+0.5x) & & y'=y(-1+x)

\end{array}$**Mon Nov 12**: Fourth Midterm. Acceleration-velocity and Population models**Wed Nov 14**: 1.4. Applications of Torricelli's Law [p.44 #54--65]**Fri Nov 16**: 3.4. Mechanical vibrations: Free undamped motion [p.195 #1--4 and if you are brave, try 10,11]**Mon Nov 19**: 3.4. Mechanical vibrations: Free damped motion [p.195 #13--23]**Mon Nov 26**: 3.6. Mechanical vibrations: Undamped forced oscillations [p.222 #1--6]**Wed Nov 28**: 3.6. Mechanical vibrations: Damped forced oscillations [at this point, you should be able to solve all problems in page 222]**Fri Nov 30**: 3.7. Electrical circuits [p.231 #1--10]**Mon Dec 03**: 1.5 & 5.2. Single and multiple tank mixture problems [p.56 #36--40; p.316 #27-37]

**Final Stretch:****Wed Dec 05**: Review**Fri Dec 07**: Review**Wed Dec 12**: 12.30PM--3:00PM

Final Exam. Chapters 1, 2, 3, 4 and 7.