Ordinary Differential Equations Homework Solutions

Math 119B: Ordinary Differential Equations
UC Davis, Spring 2012

News and announcements

  • (6/12) Have a great summer vacation!
  • (6/12) The final grades have been submitted to the registrar and should be visible through your SIS account. I have also set up a Gradebook on the course SmartSite where you can see your final exam score and the final numerical score computed as a weighted average of the different course components.
  • (6/12) Click here for the solutions to the final exam.
  • (6/6) Click here for an information sheet with details about the final exam and some tips for studying.
  • (5/16) Click here for solutions to the second midterm quiz.
  • (5/15) Click here for solutions to the review problems from HW #5.
  • (4/25) Click here for solutions to the first midterm quiz.
  • (4/10) Starting tomorrow, all lectures will take place in Wellman 107.

General course information

  • Instructor: Dan Romik
  • Lecture times and place: MWF 1:10-2:00 at Wellman 107
  • Office hours: W 2:30-3:30 in my office, MSB 2218
  • Prerequisites: MAT 21-22 lower division math series; 119A
  • Course description: Click here for a detailed syllabus.
  • Textbook and lecture notes:
    • The course will use these lecture notes (last updated on 5/29/12).
    • Part of the course will cover topics from Chapters 9 and 10 in the book Nonlinear Dynamics and Chaos, by Steven H. Strogatz.
  • Grading: The final grade will be determined from the following components:
    • Homework (the highest 4 scores from 8 weekly assignments) -- 20%
    • Two in-class midterm quizzes (25 minutes each) -- 30%
    • Final exam -- 50%
    Final and midterm dates:
    • Midterm quiz 1 -- Wednesday, April 25
    • Midterm quiz 2 -- Wednesday, May 16
    • Final exam -- Tuesday June 12, 1-3 PM

Math 1270:Ordinary Differential Equations 1Spring 2013

Instructor:David Swigon

Office: Thackeray 511, 412-624-4689, swigon@pitt.edu

Lectures:  MWF 12:00-12:50pm, Thackeray 524

Office Hours: MW 1:30-3:00pm, Thackeray 511, or by appointment.

Course Web Page: (check frequently for changes and updates) http://www.math.pitt.edu/~swigon/math1270.html

Course Description

This course is covers the theory of ordinary differential equations (ODEs) at the undergraduate level.The topics include classical methods of solving first order ODEs, linear higher-order ODEs and systems of first order linear and nonlinear ODEs. Geometric qualitative methods for autonomous systems of first order ODEs and solution of boundary value problems will be briefly discussed.


Single-variable calculus (Math 0420, 0450, or equivalent) including Taylor series, linear algebra (Math 1180, 1185 or equivalent) including eigenvalues and eigenvectors.


W.E. Boyce & R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 9th edition,John Wiley & Sons, 2009. ISBN 9780470383346 (Available at The Book Center on Fifth Ave.)

Alternative/supplementary texts:

M. Tenenbaum & H. Pollard, Ordinary Differential Equations, Dover 1985, ISBN 0486649407

E. A. Coddington, An Introduction to Ordinary Differential Equations, Dover 1989, ISBN 0486659429

Grading Scheme

Homework Assignments: 25%

Midterm Exams: 20% each

Final Exam: 35%


Homework is due at the beginning of a class one week after it was assigned. You may work with other students on homework but do not submit solutions that are identical copies of each other, as such will be discarded. 

Some homework assignments will require the use of a computer software to perform numerical computation or symbolic manipulation. You may use any computer program (e.g., XPP, ODE Architect, MATLAB, Maple, Mathematica)in solving the homework. Each such use should be explicitly documented (e.g., “eigenvalues computed with MATLAB”).A list of appropriate computer programs is given below.





Numerical solution of ODEs with GUI

Free for download, Windows and Unix versions

ODE Architect

Numerical solution of ODEs with GUI

Free with the Textbook


Numerical solution of ODEs, Symbolic manipulation, Matrix algebra

Benedum and GSCC computer labs, Windows/OSX/Linux/Unix version for $10 from Pitt SLS


Numerical solution of ODEs, Symbolic manipulation, Matrix algebra

Benedum and GSCC computer labs


Numerical solution of ODEs, Symbolic manipulation, Matrix algebra

Benedum and GSCC computer labs, Windows/OSX/Linux/Unix version for $10 from Pitt SLS

Disability Resource Services

If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both your instructor and Disability Resources and Services, 140 William Pitt Union, 412-648-7890 or 412-383-7355 (TTY) as early as possible in the term. DRS will verify your disability and determine reasonable accommodations for this course.

Academic Integrity

Cheating/plagiarism will not be tolerated. Students suspected of violating the University of Pittsburgh Policy on Academic Integrity will incur a minimum sanction of a zero score for the quiz, exam or paper in question. Additional sanctions may be imposed, depending on the severity of the infraction.On homework, you may work with other students or use library resources, but each student must write up his or her solutions independently. Copying solutions from other students will be considered cheating, and handled accordingly.







Jan 7 – Jan 11


2.1, 2.4

Introduction, Classification, First order ODEs – linear equations, Existence and uniqueness of solutions of linear equations

HW #1


Jan 14 – Jan 18

2.2, 2.4, 2.3

Separable equations, Existence and uniqueness of solutions of nonlinear equations, Models using 1st order ODEs,

HW #2


Jan 23 – Jan 25

2.5, 2.6 2.8

Autonomous equations and population dynamics, Exact equations and integrating factors

HW #3


Jan 28 – Feb 1

2.7, 8.1-3

Numerical solution of ODEs – Euler method, Improved Euler method, Error & Stabillity

HW #4


Feb 4 – Feb 8

3.1, 3.4-5

Higher order ODEs – Homogeneous equations with constant coefficients, Complex roots, Repeated roots, Fundamental solutions of linear homogeneous equations, Linear Independence and the Wronskian,

HW #5


Feb 11 – Feb 15

3.2-3, 3.6-7

Nonhomogeneous equations – undetermined coefficients, Variation of parameters

HW #6


Feb 18 – Feb 22



Feb 25


Feb 27

Midterm Exam I

Exam review sheet

Mar 1


Review of matrix theory

Mar 4 – Mar 8


Systems of first order linear ODEs, Homogeneous systems with constant coefficients Repeated eigenvalues, Complex eigenvalues, Fundamental matrices

HW #7


Mar 11 – Mar 15


Mar 18 – Mar 22

7.7, 7.1, 7.9

Matrix exponential, Nonhomogeneous linear systems, RLC circuits

HW #8


Mar 25 – Mar 29


Nonlinear differential equations, phase plane analysis

HW #9


Apr 1 – Apr 5


Stability, Cooperative systems

Apr 8

Apr 10


Midterm Exam II

Exam II review sheet

Apr 12


Predator-prey equations

Apr 15 – Apr 19


Lyapunov’s method


Saturday, Apr 27

2:00 - 3:50pm

Final Exam

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