Math 302 Differential Equations with Applications
Spring 2019

Course Information

  • Jonas Lührmann
  • Email: luehrmann (at)
  • Office: Krieger 219
  • Office hours: MF 14:30-15:30
  • MWF 12:00 - 12:50 at Mergenthaler 111
  • MWF 13:30 - 14:20 at Mergenthaler 111
  • Differential Equations with Boundary Value Problems (10th Edition)
    William E. Boyce and Richard C. DiPrima
    ISBN: 9781119925064

We will basically cover the material detailed in the official 110.302 Differential Equations Course Syllabus. However, the lectures will not follow the text verbatim and I strongly recommend to take notes in class.


There will be two midterm exams and a final exam:

  • 1st midterm exam: Monday, March 4, in class
  • 2nd midterm exam: Monday, April 15, in class
  • Final exam: Thursday, May 9, 9:00-12:00

Exams are closed book, closed notes. There will be no make-up exams. For excused absences, the grade for a missed exam will be calculated based on your performance on all remaining exams. To be excused from an exam, you have to provide documentation and a valid excuse. Unexcused absences count as 0.

Grade Policy:

The course grade will be determined as follows:
  • Homework: 10%
  • Midterm exams: 25% each
  • Final exam: 40%

Homework based on the week's lectures will be posted as official in the course schedule below, usually sometime on Friday. That assignment will be due at the beginning of class the next Friday. Hand your homework set into the bin corresponding to your section. You will receive your graded homework back from your section teaching assistant the following week. No late homework will be accepted. If you absolutely cannot make it to class, arrange for someone else to hand it in for you. However, you may miss up to two homework assignments without grade penalty, as the lowest two homework scores will be dropped from the final grade calculation.

In order to master the material of the course, it is key that you do your homework. You should make every effort to solve the assigned problems using the concepts learned from the lectures and readings. You will be graded mostly on your ability to work problems on exams. If you have not practiced the techniques within the homework problems, you will have serious difficulties to work problems on exams. You are strongly encouraged to do your homework in groups. However, you must write up your solutions on your own. Copying is not acceptable.

You must staple your homework and write your name, your section number and the name of your teaching assistant clearly at the top. Write legibly. The grader might choose not to grade your homework if it is too messy. Your solutions to the assigned problems should be detailed enough so that the reader can follow your thought process.

Course Policy:

You are responsible for lecture notes, any course material handed out, and attendance in class. I will not formally record your attendance, but you are encouraged to come to lectures. By attending lectures you will get a sense of what I consider important and that should help you know what to focus on when you study for the exams.

No cell phones and no computers are allowed during the lecture, except for note taking.

Academic Support:

Besides attending the lectures and the recitation sections I encourage you to use the following opportunities for additional academic support:
  • Come to my office hours and to your section's teaching assistant's office hours.
  • Go to the math helproom in Krieger 213. The hours are 9:00-21:00 on Monday through Thursday and 9:00-17:00 on Friday. This free service is a very valuable way to get one-on-one help on the current material of the class from other students outside the course. It is staffed by graduate students and advanced undergraduates.
  • Participate in the PILOT learning program. It is a peer-led-team learning program. Students are organized into study teams consisting of 6-10 members who meet weekly to work problems together. A trained student leader acts as captain and facilitates the meetings.
  • Make use of the Learning Den Tutoring Services.
Check out the following JHU webpage with information about academic support and tutoring.

Special Aid:
Students with disabilities who may need special arrangements within this course must first register with the Office of Academic Advising. I will need to have received confirmation from the Office of Academic Advising. To arrange for testing accomodations please remind me at least 7 days before each of the midterms or final exam by email.

JHU Ethics Statement:

The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Cheating is wrong. Cheating hurts our community by undermining academic integrity, creating mistrust, and fostering unfair competition. The university will punish cheaters with failure on an assignment, failure in a course, permanent transcript notation, suspension, and/or expulsion. Offenses may be reported to medical, law, or other professional or graduate schools when a cheater applies.

Violations can include cheating on exams, plagiarism, reuse of assignments without permission, improper use of the internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition. Ignorance of these rules is not an excuse.

In this course, as in many math courses, working in groups to study particular problems and discuss theory is strongly encouraged. Your ability to talk mathematics is of particular importance to your general understanding of mathematics. You should collaborate with other students in this course on the general construction of homework assignment problems. However, you must write up the solutions to these homework problems individually and separately. If there is any question as to what this statement means, please ask the instructor.

ODE Applets:

Here are several links to webpages with applets that are helpful for understanding the behavior of solutions to ordinary differential equations (ODEs): I find this slope field plotter very good. The following mathlet for illustrating linear phase portraits is very helpful and you may want to check out other MIT Mathlets. You can also use Wolfram Alpha to plot direction fields for systems of ODEs (enter "streamplot"). The following applet by Bluffton University is also very easy to use to plot direction fields (click on the tab "System"). Finally, there is a collection of helpful Java Applets for ODEs (JOde) from Hopkins.


Section #
Office hours

T 13:30-14:20

Bloomberg 176

Lili He

Th 14:30-16:30
Krieger 211


T 15:00-15:50

Latrobe 107

Dan Ginsberg

T 16:00-17:00
Krieger 202


Th 15:00-15:50

Gilman 55

Xiaoqi Huang

W 11:00-13:00
Krieger 200


Th 16:30-17:20

Croft Hall G02

Rong Tang

F 15:00-17:00
Krieger 211


Th 13:30-14:20

Shriver Hall 104

Lili He

Th 14:30-16:30
Krieger 211


Th 15:00-15:50

Bloomberg 274

Rong Tang

F 15:00-17:00
Krieger 211


T 16:30-17:20

Croft Hall G02

Julia Costacurta

Th 15:00-16:00
Krieger 207


Th 13:30-14:20

Shaffer 2

Xiaoqi Huang

W 11:00-13:00
Krieger 200

Course Schedule

Here is a tentative schedule for the course. It will be updated as we go with lecture notes and homework assignments. The lecture notes are only meant to supplement your own note taking in class and your reading of the textbook. Solutions to selected homework problems will be provided. I strongly recommend to you to read the relevant sections of the textbook before and/or after each lecture.

Topics and Sections

Jan 28, 30,
Feb 1

§1.1 Mathematical Models and Slope Fields
§1.2 Solutions to Some Differential Equations
§1.3 Classification of Differential Equations
§2.1 Linear Equations and Integrating Factors

Jan 28 Jan 30 Feb 1

Please become familiar with the organization of this course by carefully reading the syllabus on this webpage.

Do the following exercises from the textbook:
§1.1: 2, 4, 15-20
§1.2: 3, 7, 13
§1.3: 4, 9
§2.1: 2, 10, 20
(Use a computer to draw direction fields and print them out)

Selected Solutions

Feb 8

Feb 4, 6, 8

§2.2 Separable Equations
§2.3 Modeling with First Order Equations
§2.4 Linear vs. Nonlinear Equations
§2.5 Autonomous Equations and Population Dynamics

Feb 4 Feb 6 Feb 8

Do the following exercises:
§2.1: 22, 28, 30, 35
§2.2: 2, 5, 14, 22, 24
§2.4: 4, 5, 13, 15, 26, 27
§2.5: 2, 4, 7, 9

Read section §2.3 from the textbook

Selected Solutions

Feb 15

Feb 11, 13, 15

§2.5 Exercises: Bifurcation Theory and Diagrams
§2.6 Exact Equations and Integrating Factors
§2.8 Existence and Uniqueness Theorem

Feb 11 Feb 13 Feb 15

Do the following exercises:
§2.5: 16, 17, 26, 27
§2.6: 4, 5, 12, 14, 16
§3.1: 7, 14, 18, 20, 21, 24

Selected Solutions

Feb 22

Feb 18, 20, 22

Wednesday lecture cancelled due to weather

§3.1 Homogeneous Equations
§3.2 The Wronskian

Feb 18 Feb 22

Do the following exercises:
§3.2: 4, 11, 14, 18, 26, 28, 31, 33, 37

Selected Solutions

Mar 1

Feb 25, 27
Mar 1

§3.2 The Wronskian
§3.3 Char. Eqn. Roots: Complex
§3.4 Char. Eqn. Roots: Repeated

Feb 25 Feb 27 Mar 1

Do the following exercises:
§3.3: 3, 14, 19, 25, 27
§3.4: 8, 18, 24, 27

Selected Solutions

Mar 8

Mar 4, 6, 8

1st midterm on Monday in class

§3.5 Nonhomogeneous Equations
§3.6 Variations of Parameters
§3.7 Mech. and Electr. Vibrations

Mar 6 Mar 8

Do the following exercises:
§3.5: 3, 10, 12, 14, 17, 30
§3.6: 3, 7, 10, 15, 18, 28, 29

Read section §3.7 from the textbook.

Selected Solutions

Mar 15

Mar 11, 13, 15

§4.1 Higher Order Linear Equations
§4.2 Homogeneous Equations
§4.3 Undetermined Coefficients
§7.1 Introduction to Systems
§7.2 Review of Matrices
§7.3 Linear Algebraic Equations

Mar 11 Mar 13 Mar 15

Review Linear Algebra

Do the following exercises:
§4.1: 6, 8, 18, 19
§4.2: 11, 12, 18, 31
§4.3: 1, 2, 6, 14
§7.1: 4, 6, 15

Read sections §7.2 and §7.3 from the textbook to review linear algebra.

To supplement your review of linear algebra I suggest to do the following exercises:
§7.2: 2, 11, 23, 24
§7.3: 1, 8, 19, 20
(This is voluntary! You do not have to hand in solutions to the above exercises from §7.2 and §7.3)

Selected Solutions

Mar 29

Mar 18-24

Spring break

No homework

Mar 25, 27, 29

§7.4 First Order Linear Systems
§7.5 Homogeneous Linear Systems

Mar 25 Mar 27 Mar 29

Do the following exercises:
§7.4: 4, 6
§7.5: 4, 5, 8, 15, 16, 25, 27, 29, 31

(Use an applet to draw the direction fields and print them out)

Selected Solutions

Apr 5

Apr 1, 3, 5

§7.6 Complex Eigenvalues
§7.7 Fundamental Matrices
§7.8 Repeated Eigenvalues

Apr 1 Apr 3 Apr 5

Do the following exercises:
§7.6: 3, 9, 13, 15, 28
§7.7: 5, 8, 11
§7.8: 4, 8, 15

Selected Solutions

Apr 12

Apr 8, 10, 12

§9.1 The Phase Plane
§9.2 Autonomous Systems and Stability
§9.3 Locally Linear Systems

Apr 8 Apr 10 Apr 12

Do the following exercises:
§9.1: 15, 20, 21
§9.2: 5, 8, 18

Apr 19

Apr 15, 17, 19

2nd midterm on Monday in class

§9.3 Locally Linear Systems
§9.4 Competing Species

Apr 17 Apr 19

Do the following exercises:
§9.3: 5, 16, 20, 27
§9.4: 2, 17

Apr 26

Apr 22, 24, 26

§9.5 Predator-Prey Equations
§9.7 Periodic Solutions and Limit Cycles
§6.1 Definition of the Laplace Transform

Apr 29
May 1, 3

§6.2 Solution of Initial Value Problems
§6.3 Step Functions
§6.4 Discontinuous Forcing Functions
§8.1 The Euler or Tangent Line Method
Course Review

Final exam:

Thursday, May 9, 9:00-12:00


Tue, Apr 16: Here are the solutions to our second midterm exam.

Fri, Apr 5: Here is a study guide for our second midterm exam. It includes practice problems that I recommend you to do as part of your preparation for the exam. And here are the solutions to the practice problems.

Fri, Apr 5: Please read this note on the logistics of our second midterm exam on Monday, 15 April 2019; it is the same logistics as for our first midterm exam. Make sure you know in which room and at which time you will be taking the exam.

Tue, Mar 5: Here are the solutions to our first midterm exam: Solutions Group A and Solutions Group B.

Wed, Feb 27: Please carefully read this note on the logistics of our first midterm exam on Monday, 4 March 2019. Make sure you know in which room and at which time you will be taking the exam.

Wed, Feb 20: Here is a study guide for our first midterm exam. It includes practice problems that I recommend you to do as part of your preparation for the exam. And here are the solutions to the practice problems.

Mon, Jan 28: Welcome to Math302 Differential Equations! I wish you all the best for this spring term.