# MATH 242 Fall 2018. Section 001

## Meeting Times

Section Schedule Location
001 MWF 09:40 AM - 10:30 AM LeConte 112

## 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
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

• In-class tests:
There will be four in-class 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 in-class 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.

The course grade will be determined as follows:

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, 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]
• 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 one-on-one appointment with a peer tutor by going to www.sc.edu/success. Drop-in 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.
SSC services are offered to all USC undergraduates at no additional cost. You are invited to call the Student Success Hotline at (803) 777-1000 or visit www.sc.edu/success to check schedules and make appointments. Success Consultants are available to assist you in navigating the University and connecting to available resources.

## 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
• 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.

## Lesson Plan

### Classical Methods

• Fri Aug 24

General Introduction to Differential Equations

[p.8 #1--26]

• Mon Aug 27 [Review: Integration]

Integrals as general and particular solutions

[p.15 #1--10]

• Wed Aug 29

—Slope fields and numerical approximation

—Euler's method

[p.25 #1--10; p.113 #1,4,6,10]

• Fri Aug 31 [slides]

Separable equations

Singular Solutions

[p.40 #1--28]

• Wed Sep 05 [slides]

Homogeneous equations

[p.69 #2,3,7--10,12--14]

• Fri Sep 07 [slides]

Linear first-order differential equations

Bernoulli equation

[p.53 #1--21 and the equations below]

1. $$xy' +y = y^2\ln x$$
2. $$y'+y \dfrac{x+\tfrac{1}{2}}{x^2+x+1} = \dfrac{(1-x^2)y^2}{(x^2+x+1)^{3/2}}$$
3. $$(1+x^2) y' =xy+x^2y^2$$
4. $$x^2y'+2x^3y=y^2(1+2x^2)$$
5. $$3y'+y \dfrac{x^2+a^2}{x(x^2-a^2)} = \dfrac{1}{y^2} \dfrac{x(3x^2-a^2)}{x^2-a^2}$$
6. $$y' + \dfrac{y}{x+1} = -\frac{1}{2} (x+1)^3 y^2$$

• Mon Sep 10 [slides]

General substitution methods

[p.69 #1,4--6,15--18]

• Wed Sep 12 [slides]

Exact equations

[p.69 #31--42]

• Fri Sep 14 [slides]

Reducible Second-order Differential Equations

[p.69 #43--54]

• Mon Sep 17 [slides]

Intro to second-order linear differential equations

[p.147 #1--16]

• Wed Sep 19 [slides]

Homogeneous linear second-order differential equations with constant coefficients

[p.170 #33--42]

• Fri Sep 21 [slides]

Particular solutions for Second-order linear differential equations with constant coefficients

—The method of variation of parameters.

[p.195 #1--20, 47--56] Use exclusively the method of variation of parameters

• Mon Sep 24 [slides]

Particular solutions for Second-order linear differential equations with constant coefficients

—The method of undetermined coefficients (Part I: the easy examples)

[No HW today]

• Wed Sep 26 [slides]   [Recording of the review session]   [Transcript (pdf)]

Particular solutions for Second-order 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 #1--56 in p.210 using either method]

• Fri Sep 28

General solutions to Second-order linear differential equations with constant coefficients

Review

• Mon Oct 01

Classes Canceled

• Wed Oct 03

First Test. Classical Methods

### Methods based on the Laplace Transform

• Fri Oct 05 [slide | slide]

Improper integrals revisited

[p.445 #1--6]
Find the Laplace transform of $$f(x) = \cos \beta x$$, and $$f(x) = 1/\sqrt{x}$$ using the definition.

• Mon Oct 08 [slides]

—Linearization of Transforms

—Translation on the s-axis

[p.446 #13,16--21,23,26--32,35; p457 #28--31; p.464 #1--22]

• Wed Oct 10 [slides]

—Differentiation of Transforms

Use the table of transforms to find the Inverse Laplace Transform of the following functions:

1. $$F(s) = \dfrac{3}{s^4}, (s>0)$$
2. $$F(s) = \dfrac{5}{s+5}, (s>-5)$$
3. $$F(s) = \dfrac{3}{s-4}, (s>4)$$
4. $$F(s) = \dfrac{3s+1}{s^2+4}, (s>0)$$
5. $$F(s) = \dfrac{5-3s}{s^2+9}, (s>0)$$
6. $$F(s) = \dfrac{9+s}{4-s^2}, (s>2)$$
7. $$F(s) = \dfrac{1}{s(s-3)}, (s>3)$$
8. $$F(s) = \dfrac{3}{s(s+5)}, (s>0)$$
9. $$F(s) = \dfrac{1}{s(s^2+4)}, (s>0)$$
10. $$F(s) = \dfrac{2s+1}{s(s^2+9)} , (s>0)$$
11. $$F(s) = \dfrac{1}{s(s^2-9)}, (s>3)$$
12. $$F(s) = \dfrac{1}{s(s+1)(s+2)}, (s>0)$$
13. $$F(s) = \dfrac{2(s-4)+3}{(s-4)^2+25}), (s>4)$$
14. $$F(s) = \dfrac{5s-6}{s^2-3s}, (s>3)$$
15. $$F(s) = \dfrac{5s-4}{s^3-s^2-2s}, (s>2)$$
16. $$F(s) = \dfrac{1}{s^4-16}, (s>2)$$
Find the Laplace Transform of the following functions:
1. $$f(x) = x^4 e^{\pi x}$$
2. $$f(x) = e^{-2x} \sin (3\pi x)$$

• Fri Oct 12 [slides]

—Laplace transform of derivatives

—Transformation of Initial Value Problems

• Mon Oct 15

• 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.82 #9--12, 21--24]

• Mon Oct 29

Population models

—Equilibrium solutions and stability

[p.91 #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 Oct 31

Review

• Fri Nov 02

Third Test. Geometric Applications and Population Models

• Mon Nov 05

Acceleration-velocity models (Part I)

[p.15 #24--29,33,37]

• Wed Nov 07

Acceleration-velocity models (Part II)

[p.100 #7--10,17--20]

• Fri Nov 09

Mechanical vibrations

—Free undamped motion

[p.181 #1--4 and if you are brave, try 10,11]

• Mon Nov 12

Mechanical vibrations

—Free damped motion

[p.181 #13--23]

• Wed Nov 14

Mechanical vibrations

—Undamped forced oscillations

[p.206 #1--6]

• Fri Nov 16

Mechanical vibrations

—Damped forced oscillations

Electrical circuits

[at this point, you should be able to solve all problems in page 206. p.231 #1--10]

• Mon Nov 19

Fourth Test. Acceleration/Velocity and Mechanical Vibrations

• Mon Nov 26

Applications of Torricelli's Law

[p.44 #54--65]

### Reviews

• Wed Nov 28 [Review (1|5)]

Integration skills

• Fri Nov 30 [Review (2|5)]

Classical Methods

• Mon Dec 03 [Review (3|5)]

Methods based on Laplace Transform

• Wed Dec 05 [Review (4|5)]

Geometric Applications, Population Models

• Fri Dec 07 [Review (5|5)]

Acceleration/Velocity, Mechanical Vibrations, Torricelli