# Differential equations course webpage

**Administrative**

- SM212 policy statement, SM222 policy statement
- Syllabus for Spring 2016-2017, SM212
- Exercises for an experimental section of SM212P
- course objectives.
- How to start computer project 1 using SageMath in the cloud

**SAGE examples**

All SAGE code (if any) in the notes is licensed under the GPL, version 2 or greater (your choice). You can access Sage at the sagemath cloud server.

- Newton's Law of Cooling CP.
- Euler's method, improved Euler's method, and direction fields using SAGE, and in pdf.
- the motion of springs with large displacements CP.
- Euler's method for 2x2 systems CP
- Sage and DEs. - short article written November 2008 for Mathematica Militaris with former students, N. Albertson, C. Miller and N. Peters.
- Love, War and Zombies (slides), Sage code

**Sample Tests and quizzes**

- Test 1, 1997 (with solutions)
- Test 1, 1998 (with solutions).
- Quiz 1 (with solutions).
- Quiz 2 (with solutions).
- team quiz on springs (no damping) (with solutions).
- Practice Test 2 pdf on constant coefficient ODEs, undetermined coefficients,

springs, circuits, and variation of parameters (with solutions). - review for test 2 written by Prof G. Nakos (a pdf file)
- Test 3 on systems, circuits, and Laplace transforms (with solutions).
- practice test 3 on systems, circuits, and Laplace transforms (with solutions)
- Spring 2009 Test 3 from Prof Gaglione with complete solutions, on systems, circuits, matrices and eigenvalues,
- practice test 4 on matrices, systems by eigenvalues, sine/cosine/Fourier series,

separation of variables for PDEs and the heat equation. (with solutions) - An old Test 4 on Fourier series and PDEs. Another old Test 4. And another old Test 4.
- Fall 1988 final in pdf. (1(b), 6(b), and 8 are not on this semester's syllabus, so you should skip them.

Also, the diagram for 7(b) is missing, so you can skip that since it is incomplete.)

Solutions (pdf). - Fall 1991 final, in pdf. (1(c), 6(b), 9(a), and 9(b) are not on this semester's syllabus, so you can skip them.
- Fall 1996 final, in pdf. With solutions. (1(a)(ii) and 6(b) are not on this semester's syllabus, so you can skip them.)
- Practice exam (Spring 2002) Multiple choice (pdf), Written portion (pdf). With answers.

(5, 14, 16 and 17 on the MC portion are not on this semester's syllabus, so you should skip them. ) - Final exam (Fall 2003) pdf. With solutions.
- Final exam (Spring 2004) written pdf. multiple choice pdf. solutions pdf. (4 is not on this semester's syllabus, so you can skip it.)
- Fall 2009-2010 final, Spring 2008-2009 final (pdfs).
- Spring 2016-2017 final, errata (one minor typo is corrected), solutions. Note: Due to the transition in calculators, this was a no calculator final. The "mini tables" at the bottom of this page were handed out with the final.

**Lecture notes and handouts (pdf files)**

The files marked "(with SAGE examples)" also have source tarballs available (replace "de-*.pdf" by "de-*.tar.gz") and both are licensed under the Attribution-ShareAlike Creative Commons license. All SAGE code (if any) in the notes is licensed under the GPL, version 2 or greater (your choice).

- Partial fractions handout
- Introduction to matrix determinants handout
- Lanchester's model handout
- Impulse-response handout
- Fourier series handout
- Heat equation (PDE) handout
- Introduction to ODEs (with SAGE examples)
- Initial value problems (with SAGE examples)
- Existence and uniqueness (with SAGE examples). picard iteration Sage code.
- Euler's method for numerically approximating solutions to DEs.

Includes both 1st order DE case (with Euler and improved Euler) and higher order DE and systems of DEs cases, without improved Euler (with SAGE examples). - Direction fields and isoclines (with SageMath examples)
- 1st order ODEs, separable and linear cases (with SAGE examples)
- A falling body problem in Newtonian mechanics. (with SAGE examples)
- A mixing problem. (with SAGE examples)
- Linear ODEs, I (with SAGE examples)
- Linear ODEs, II (with SAGE examples)
- Undetermined coefficients for non-homogeneous 2nd order constant coefficient ODEs. (with SAGE examples)
- Variation of parameters for non-homogeneous 2nd order constant coefficient ODEs. (with SAGE examples)
- Annihilator method for non-homogeneous 2nd order constant coefficient ODEs.
- Springs, I (with SAGE examples)
- Springs, II (with SAGE examples)
- Springs, III (with SAGE examples)
- LRC circuits (with SAGE examples)
- Power series methods, I
- Power series methods, II (with SAGE examples)
- Introduction to Laplace transform methods, I (with SAGE examples)
- Introduction to Laplace transform methods, II (with SAGE examples)
- Lanchester's equations modeling the battle between two armies. (with SAGE examples), video.
- Row reduction/Gauss elimination method for systems of linear equations. (with SAGE examples)
- Eigenvalue method for homogeneous constant coefficient 2x2 systems of 1st order ODEs. (with SAGE examples)
- Variation of parameters for first order non-homogeneous linear constant coefficient
*systems*of ODEs. - Electrical networks using Laplace transforms (with SAGE examples)
- Separation of variables and the Transport PDE (with SAGE examples).
- Fourier series (with SAGE examples).
- one-dimensional heat equation using Fourier series (with SAGE examples).
- one-dimensional wave equation using Fourier series.
- one-dimensional Schroedinger's wave equation for a "free particle in a box" using Fourier series.
- All these lectures collected as one pdf (216 pages). The latest version, written with Marshall Hampton, is this pdf (197 pages). Licensed using the open source Attribution-ShareAlike Creative Commons license.

The individual lectures were updated on 2008-12-01 but the 216 collection has not yet been updated and still has several typos. Please use the 197 page collection, which is the most recent of all.

Course review: pdf

## Useful links

- Robert Marik and Miroslava Tihlarikova's
*Mathematical Assistant*is a cool program which solves many 1st or 2nd order ODEs and gives you the complete solution (with all the steps) as a pdf. You can enter the ODE yourself into a form interface on his webpage. If you'd like to try it out, Please refer to the Differential equations course webpage

- Sage and direction fields and phase portraits, tutorial/paper by Prof Corso (posted with his kind permission).
- The differential equations course at MIT's "open courseware" site. Includes lecture notes, exercises, and videotaped lectures (videos require RealPlayer). Some of the pdfs on the MIT website are hard to print out so some of them have been converted and posted locally.
- The excellent lecture notes and examples of Prof. Paul Dawkins (also available as a free pdf).
- Prof. Trench's excellent book Elementary Differential Equations with Boundary Values (free pdf).
- Prof. Lebl's Notes on Diffy Qs (free pdf).
- Regarding the 2-dim wave equation: "Can One Hear the Shape of a Drum?"
- Excellent lectures and videos on many areas of engineering mathematics, including some differential equations, can be found at the numerical methods website.
- 9 Wonderful book,
**Sage for Undergraduates,**with a section on applications of Sage to DEs. The pdf is free.

## Math Tables

These are in the public domain.

New: 2006 version of the USNA Math Tables (pdf file - missing the probability tables, but with more calc+Precalculus info)

Same tables (pdf - but reduced 2-pages-to-a-page).

Laplace transform tables only (pdf file)

Laplace transform tables only (another version using newer fonts, pdf file)

Latex: source

Mini-tables: pdf.