# MATH 242 Summer 2017

## Section 202

Schedule Location
MTWTh 1:45 PM - 3:10 PM LeConte 405

## Important deadlines you need to know

General Dates
Part of term begins June 26, 2017
Part of term ends August 10, 2017
Last Day to Drop/Add June 28, 2017
Last Day to Change Credit/Audit June 28, 2017
First Day 'W' Grade Assigned June 29, 2017
Last Day 'W' Grade Assigned July 18, 2017
Last Day to Elect Pass/Fail July 18, 2017
First Day 'W/F' Grade Assigned July 19, 2017

## Prerequisites

A grade of C or better in MATH 142

## Course Structure and Grading Policies

• In-class tests:
There will be five in-class tests scheduled as follows:
Test # Date
1 Wed, Jul 05
2 Thu, Jul 13
3 Mon, Jul 24
4 Mon, Jul 31
5 Mon, Aug 07
• Final Exam:
If you have taken at least three of the in-class exams, and are unhappy with the final score obtained from the in-class exams, notify me by email on Thursday, Aug 10, before 8:00 PM. You will have an opportunity to change your course grade by taking a (comprehensive) final exam on Saturday, August 12th, at 12:30 PM.

The course grade will be determined as follows:

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, 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

• First part: Introduction to Differential Equations
• Mon Jun 26: 1.1, 1.2. [Review: Integration]
General Introduction to Differential Equations.
Integrals as general and particular solutions.
[p.8 #1--31,37--42, p.15 #1--10]
• Tue Jun 27: 1.3, 2.4, 2.5.
Slope fields.
Numerical Approximation: Euler's method.
[p.24 #1--10, p.113 #1,4,6,10]
• Wed Jun 28: 1.4, 1.6. [slide | slide | slide]
Separable equations. Singular Solutions.
Homogeneous First-Order equations.
[p.40 #1--28, p.69 #2,3,7--10,12--14]
• Thu Jun 29: 1.5, 1.6. [slides]
Linear first-order differential equations.
[p.53 #1--25]
• Wed Jul 05: First Test
• Thu Jul 06: 3.1, 3.2, 3.3. [slides]
Intro to linear differential equations of Higher Order.
Second-Order Linear Equations.
[p.147 #1--16]
• Mon Jul 10: 3.5. [slide | slide | slide]
Homogeneous linear second-order differential equations with constant coefficients.
Particular solutions for Second-order linear differential equations with constant coefficients:
- Variation of parameters.
[p.147 #33--42, p.170 #1--9,21,22,23, p.195 #1,2,3,6,10]
• Tue Jul 11: 3.5. [slide]
Particular solutions for Second-order linear differential equations with constant coefficients:
- Undetermined coefficients.
[p.195 #21,23,26,31--35,53--55]
• Wed Jul 12: Review
• Thu Jul 13: Second Test
• Second Part: Laplace Transform
• Mon Jul 17: 7.1, 7.2. [slides]
Improper integrals revisited.
[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 Jul 18: 7.4, 7.3 [slide | slide | slide]
Linearization and Translation on the s-axis.
Assignment
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. $$f(x) = e^{-2x} \sin (3\pi x)$$
• Wed Jul 19: 7.2. [slide]
Differentiation of Transforms.
Laplace transform of derivatives.
Transformation of Initial Value Problems.
[p.456 #1--10, p.464 27--30, p.473 #15,16,17]
• Thu Jul 20: Review
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.
• Mon Jul 24: Third Test
• Third Part: Applications to Mathematical Modeling
• Tue Jul 25: 1.2, 2.3.
Acceleration-velocity models
[p.100 #2,4,6,9,11,17,18,20]
• Wed Jul 26: 2.1, 2.2.
Population models
Equilibrium solutions and stability
[p.40 #33,34,37, p.82 #15,16,21,22,23,26]
• Thu Jul 27: Review
• Mon Jul 31: Fourth Test
• Tue Aug 01: 3.4.
Mechanical vibrations:
- Free undamped motion
- Free damped motion
[p.181 #1--4,13]
• Wed Aug 02: 3.6.
Mechanical vibrations:
- Undamped forced oscillations
- Damped forced oscillations
[p.181 #15--21, p.206 #1--4]
• Thu Aug 03: Review
• Mon Aug 07: Fifth Test
• Tue Aug 08: 3.7.
Electrical circuits
• Wed Aug 09: 1.4.
Applications of Torricelli's Law
[p.42 #54--64]
• Thu Aug 10: Review