### Meeting Times

 Lectures: MWF 12:20 AM - 1:10 PM LeConte 115 Office Hours: TTh 2.00 PM — 5.00 PM LeConte 307

### Prerequisites

Qualifications through placement or a grade of C or better in MATH 142. The deadline to drop/add is Friday, January 13th. The first day in which a "W" grade is assigned is therefore Saturday, January 14th. The last day to obtain a "W" grade or to elect a pass/fail grade is Monday, February 27th. The first day in which a "WF" grade is assigned is therefore Tuesday, February 28th.

### Text

Differential Equations: Computing and Modeling by C. Henry Edwards and David E. Penney. Prentice Hall 2008 (fourth 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 counts 20% of the final grade, for a total of 60% of the course grade. There will be three in-class midterm exams tentatively scheduled as follows:
 Test # Date 1 Wed Feb 01 2 Fri Feb 24 3 Mon Mar 26

The dates of the tests will be confirmed during the class, generally the previous week.

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, April 25th, at 2:00 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.
• 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.
• The Student Success Center and one of four Academic Centers for Excellence (ACE) are on the mezzanine level of the Thomas Cooper Library and can be reached by phone at (803) 777-0684 or by going online at [www.sc.edu/academicsuccess] Other ACE locations around campus make access to these resources easy (Sims Hall, Bates House, Columbia Hall). The centers are at the crossroads of services and information about many special resources for students, including advice on developing successful study habits, time management, and effective learning strategies as well as availability of tutoring.

### 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

• Mon Jan 09: 1.1. General Introduction to Differential Equations [p.8 #1--26]
• Wed Jan 11: 1.1 & 1.2. Intro to modeling. Integrals as general and particular solutions. [p.9 #27--36; p.17 #1--10]
• Fri Jan 13: 1.4. Separable equations and applications. [p.43 #1--28,33,37,43]
• Wed Jan 18: 1.6. Homogeneous equations. [p.74 #2,3,7--10,12--14]
• Fri Jan 20: 1.5 & 1.6. Linear first-order differential equations. Bernoulli equation [p.56 #1--21 and the equations below]
$latex \begin{array}{ll} (1)\quad xy' +y = y^2\ln x & (4)\quad x^2y'+2x^3y=y^2(1+2x^2) \\ \\ (2)\quad y'+y\displaystyle{\frac{x+\tfrac{1}{2}}{x^2+x+1}}= \displaystyle{\frac{(1-x^2)y^2}{(x^2+x+1)^{3/2}}} & (5)\quad 3y'+y\displaystyle{\frac{x^2+a^2}{x(x^2-a^2)}}=\displaystyle{\frac{1}{y^2} \frac{x(3x^2-a^2)}{x^2-a^2}} \\ \\ (3)\quad (1+x^2) y' =xy+x^2y^2 & (6)\quad y' + \displaystyle{\frac{y}{x+1}}=-\frac{1}{2} (x+1)^3 y^2\end{array}$
• Mon Jan 23: 1.6. General substitution methods. [p.74 #1,4--6,15--18,43--54]
• Wed Jan 25: 2.1. Population models [p.87 #9--12]
• Fri Jan 27: 2.2. More population models. Equilibrium solutions and stability [p.98 #1--12 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]
• Mon Jan 30: 1.3 & 2.4. Slope fields and numerical approximation. Euler's method [p.27 #1--10; p.121 #1,4,6,10]
For extra credit, code Euler's method in your favorite programming language/mathematical software. The code should accept random functions as input, initial conditions, an interval length and a step length. It should output the graph of the computed solution, or a table of the corresponding values.
• Wed Feb 01: First Midterm. Chapter 1 [Practice Exam #1]
• Fri Feb 03: How does it all fit together? Case studies [p.88 #21--24; p.98 #13--18]
• Mon Feb 06: 1.5. Improved Euler's Method [p.132 #1--10,27,28]
• Wed Feb 08: 1.2 & 2.3. Acceleration-velocity models (Part I) [p.18 #24--29,33,37,39]
• Fri Feb 10: 2.3. Acceleration-velocity models (Part II) [p.108 #7--10,17--20]
• Mon Feb 13: 3.1. Intro to Higher-order differential equations [p.158 #1--16]
• Wed Feb 15: Linear independence of solutions and Wronskians. Homogeneous linear second-order differential equations with constant coefficients [p.158 #20--26,33--42]]
• Fri Feb 17: 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
• Mon Feb 20: Second Midterm. Chapter 2 and sections 1.2 & 1.3 from Chapter 1 [Practice Exam #2]
• Wed Feb 22: 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]
• Fri Feb 24: 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]
• Mon Feb 27: 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]
• Wed Feb 29: 7.2. Laplace transform: The Gamma function. Laplace transform of derivatives. [p.462 #1--16]
• Fri Mar 02: 7.4. Laplace transform: Differentiation of Transforms [p.462 #17--22; p.481 #15,16]
• Mon Mar 12: 7.3. Laplace transform: Translation of the s-axis. The convolution property [p.472 #1--22, 27--38; p.481 #1--14]
• Wed Mar 14: 7.2. Laplace transform: Integration of Transforms. Putting it all together (transformation of Initial Value Problems)
• Fri Mar 16: 3.4. Mechanical vibrations: Free undamped motion [p.195 #1--4 and if you are brave, try 10,11]
• Mon Mar 19: 3.4. Mechanical vibrations: Free damped motion [p.195 #13--23]
• Wed Mar 21: 3.6. Mechanical vibrations: Undamped forced oscillations [p.222 #1--6]
For extra credit on the final exam (details to be discussed in class), start looking for a project. A good place to begin your search is in plus.maths.org
• Fri Mar 23: Third Midterm. Chapters 3 (minus applications) and 7 [Practice Exam #3]
• Mon Mar 26: 3.6. Mechanical vibrations: Damped forced oscillations [at this point, you should be able to solve all problems in page 222]
• Wed Mar 28: 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]
• Fri Mar 30: 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]
• Mon Apr 02: Systems of differential equations: Critical points. 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.]
$latex \begin{array}{rlrl} (1) & x'=x(1.5-x-0.5y) & (2) & x'=x(1.5-x-0.5y) \\ & y'=y(2-y-0.75x) & & y'=y(2-0.5y-1.5x) \\ \\ (3) & x'=x(1.5-0.5x-y) & (4) & x'=x(1.5-0.5x-y) \\ & y'=y(2-y-1.125x) & & y'=y(0.75-y-0.125x) \\ \\ (5) & x'=x(1-x-y) & (6) & x'=x(1-x+0.5x) \\ & y'=y(1.5-y-x) & & y'=y(2.5-1.5y+0.25x) \end{array}$
• Wed Apr 04: Systems of differential equations: Functional relationship among variables. 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 relationships between the two variables.]
$latex \begin{array}{rlrl} (1) & x'=x(1.5-0.5y) & (2) & x' = x(1-0.5y) \\ & y'=y(-0.5+x) & & y'=y(-0.25+0.5x) \\ \\ (3) & x'=x(1-0.5x-0.5y) & (4) & x'=x(1.25-x-0.5y) \\ & y'=y(-0.25+0.5x) & & y'=y(-1+x) \end{array}$
• Fri Apr 06: 3.7. Electrical circuits [p.231 #1--10]
Project drafts are due today. You are expected to have at this point both the model and the strategy to solve it.
• Mon Apr 09: 1.4. Applications of Torricelli's Law [p.44 #54--65] Junkyard wars: For extra credit, use problem 64 to build your own water clock
• Wed Apr 11: 8.1. Series solutions [p.516 #1--10]
• Fri Apr 13: 1.5 & 5.2. Single and multiple tank mixture problems [p.56 #36--40; p.316 #27-37]
Projects are due today.
• Mon Apr 16: 1.6. Conservative vector fields, and their relationship to differential equations: Exact equations [p.74 #31--42]
• Wed Apr 18: Review [Practice questions for the fourth part of the course]
• Fri Apr 20: Review
• Mon Apr 23: Review

• Wed Apr 25: 2.00pm Final Exam. Chapters 1, 2, 3, 4 and 7.