MTH 256 Applied Differential Equations

Instructor Information

Dr. Jeffrey Ovall

I have been teaching mathematics for roughly 30 years. Differential equations (MTH 256 or equivalent) is one of my favorite undergraduate courses to teach. My favorite result in this course is that any solution of a scalar autonomous equation must be monotone.

I am committed to creating an accessible learning environment for all of my students. If you encounter any accessibility barriers in this course, please reach out to me via email. I will work with you and our campus partners to evaluate the issue and find solutions.

Office: Fariborz Maseeh Hall (FMH), 464R

Office Hours: TR 1-2pm, or by special appointment

Email: jovall@pdx.edu

Phone:(503) 725-3610

Course Overview

The course introduces elementary solution techniques and methods of qualitative analysis of ordinary differential equations and systems of differential equations. It also discusses some applications in the physical and engineering sciences, e.g. mass-spring systems and population dynamics.

Prerequisites: MTH 252Z (Calculus II), MTH 261 (Introduction to Linear Algebra)

Course Meeting Times: TR 9-10:40am

Course Meeting Location: Fariborz Maseeh Hall (FMH), 200

Course Reference Number (CRN): 61450 (Section 001)

Syllabus

Learning Outcomes

Solve certain types of elementary ordinary differential equations (ODEs)
Solve systems of linear ODEs with constant coefficients
Be familiar with the concept of equilibrium solutions and their different types, and understand the relation between types of equilibrium solutions and long-term behavior of (non-equilibrium) solutions
Be familiar with the concept of the Laplace transform, and use it effectively
Effectively use these solution techniques in some applications

Textbook and Topics

We will be using the textbook Differential Equations (Fourth Edition), by Blanchard, Devaney and Hall (opens in new tab) . It is NOT required that you pay for access to the software DETools that is associated with this textbook, or any other study guides or solutions manuals. You may wish to consider e-book or rental options. A PDF of the first chapter of the book is in the Supplementary Materials section of this site.

Topics: We will cover much of Chapters 1-6, as outlined below and in the Weekly Schedule.

First-order differential equations (Ch. 1)
- Mathematical models involving first-order differential equations
- Separation-of-variables and the method of integrating factors (analytical techniques)
- Slope fields and phase lines, Bifurcations (qualitative/geometric techniques)
- Euler's method (a numerical approximation technique)
- Theory of existence and uniqueness of solutions
First-order systems of differential equations (Ch. 2, Ch. 3, Ch. 5)
- Some first-order (non-linear) system models
- Qualitative (geometric) behavior of solutions
- Analytic methods for some special systems
- Analytic techniques and qualitative behavior
- Classifying equilibria of non-linear systems, local behavior of solutions
Second-order linear differential equations (Ch. 2, Ch. 3, Ch. 4)
- The damped harmonic oscillator model, and more general second-order equations
- Analytic techniques and qualitative behavior
- Forced harmonic oscillators
- The resonance phenomenon
The Laplace Transform (Ch. 6)
- Definition, key properties, inverse
- Laplace Transforms of Discontinuous functions and Delta "functions"
- Solving first-order linear equations using Laplace Transform Methods
- Solving second-order linear equations using Laplace Transform Methods

Other topics may be covered if time permits, including: Existence/uniqueness theory for systems (Ch. 2), the SIR epidemic model (Ch. 3), the Lorenz Equations (Ch. 3, Ch. 8), Hamiltonian systems (Ch. 5), the power series approach and Bessel functions/equations (Appendix B, instructor notes), more general numerical methods beyond Euler's method (instructor notes).

Course Grade

Your course grade will be based on your best eight (out of nine) assignments, two midterm exams, and one comprehensive final exam. Your course grade will be assigned based on the percentage earned of 500 possible points:

(8x25)/2=100 for assignments
100 for each of the two midterms
200 for the final exam

Extra Credit: Do not expect extra credit or make-up work.

Assignments are due at the beginning of class on Thursdays, starting in Week 2. The problems for each assignment are listed in the Weekly Schedule. I will select four problems from each assignment for grading, each worth 5 points, and will also give up to 5 points for completeness and clarity of the submission. Your assignments should be clearly written and well-organized. It should be easy to identify and read, both your work (not "scratch work") and your final answers. Solutions to all assigned problems will be posted after the due date. In the Supplementary Materials section, there will also be solutions to some unassigned problems from the sections covered in class. For extra practice, you may wish to attempt these problems and compare your solutions with mine.

Exam Dates: Because the dates of exams are given well in advance, make/adjust your travel plans accordingly. Only in exceptional cases (left to my discretion, but including observance of religious holidays) will an exam be given on an alternate date. Unless a missed exam is due to a properly documented sickness or family emergency, an alternate exam date will be given only if we have a written agreement to do so, made at least one week before the originally-scheduled exam date.

Midterm 1: Thursday, April 16 (Week 3)
Midterm 2: Tuesday, May 12 (Week 7)
Final: Tuesday, June 9, 9-10:50am

Scores on each assignment and exam may be adjusted (in your favor) in order to achieve a fair distribution of grades. At the end of the quarter each student's points will be added and their percentage of the total points will be calculated. You are guaranteed that, if your percentage meets the traditional university standards, then you will get at least the following grade:

A: 90-100%
B: 80-89%
C: 70-79%
D: 60-69%
F: 0-59%

In assigning final grades, plusses/minuses will be used. If you have chosen P/NP grading, a grade of P will be assigned if you would have gotten at least a C under the grading scheme described above, and a grade of NP otherwise. If you are concerned about your performance at any point in the term, come talk to me---make an appointment if necessary.

Weekly Schedule

Weekly Topics and Assignments
Week	Tuesday	Thursday	Assignment (due Thursday in class)
1: Mar 31, Apr 2	Modelling via differential equations (1.1)	Separable equations (1.2)
2: Apr 7, Apr 9	Linear equations (1.8) Integrating factors (1.9)	Equilibria and phase lines (1.6) Bifurcations (1.7)	Section 1.1: 4, 11, 17, 18 Section 1.2: 5, 17, 25, 39, 41 Section 1.8: 1, 3, 5, 11
3: Apr 14, Apr 16	Slope fields (1.3) Euler's method (1.4) Existence and uniqueness of solutions (1.5)	Q&A before midterm Midterm covering Chapter 1 (except 1.4, 1.5)	Section 1.9: 1, 2, 9, 12, 24 Section 1.6: 1, 5, 13, 17, 24, 31, 34 Section 1.7: 3, 4, 13
4: Apr 21, Apr 23	Modeling via systems (2.1) Geometry of systems (2.2)	Special systems (2.4) 2x2 linear algebra review	Download Section 1.3-1.5 Worksheet(opens in new tab)
5: Apr 28, Apr 30	Overview of linear systems of ODEs (3.1-3.5) Linear systems with real, distinct eigenvalues (3.1-3.3)	Linear systems with complex eigenvalues (3.1, 3.4)	Section 2.1: 1, 2, 7a, 16a Section 2.2: 11, 12, 16ac, 18ac, 21 Section 2.4: 2, 4, 7, 10
6: May 5, May 7	Linear systems with repeated real eigenvalues (3.1, 3.2, 3.5) Trace-Determinant plane (3.7)	Equilibrium point analysis for nonlinear systems (5.1)	Section 3.1: 6, 7, 16, 24 Section 3.2: 1abde, 4abde, 14b, 19 Section 3.3: 1, 6, 12, 27abc
7: May 12, May 14	Q&A before midterm Midterm covering 2.1, 2.2, 2.4, 3.1-3.5, 5.1	Separable Equations (1.2)	Section 3.4: 2, 4abcd, 10 Section 3.5: 2, 6, 18 Section 5.1: 4, 6, 8ab, 14ab
8: May 19, May 21	Damped harmonic oscillators (2.3) Second-order linear equations (3.6)	Sinusoidal forcing (4.2) Undamped forcing and resonance (4.3)	Section 2.3: 1bc, 5, 8b. You do not have to use "guess-and-check". You may use techniques discussed in class. Section 3.6: 8, 10, 18abc
9: May 26, May 28	Laplace transforms (6.1)	Laplace transforms (6.1) Discontinuous functions (6.2)	Section 4.1: 2, 4, 6, 16 Section 4.2: 4, 8, 12 Section 4.3: 4, 10, 21
10: Jun 2, Jun 4	Second-order equations (6.3) Impulse forcing (6.4)	Optional Topic Review for final exam	Section 6.1: 8, 10, 16, 24 Section 6.2: 6, 10, 14 Section 6.3: 18, 28, 32

Supplementary Materials

The following links are either to external websites or to PDF supplements for the course.

MTH 256 Applied Differential Equations, Spring 2026