MTHSC 208, Fall 2009
# MTHSC 208, Fall 2009

"You see this little hole? This moth's just about to emerge. It's in
there right now, struggling. It's digging it's way through the thick
hide of the cocoon. Now, I could help it - take my knife, gently widen
the opening, and the moth would be free - but it would be too weak to
survive. Struggle is nature's way of strengthening it."

--Locke (*Lost*, 2004)

**Instructor**: Dr. Matthew
Macauley

**Class**: Introduction to Ordinary Differential Equations

Course Syllabus

**Week
1**: Introduction to differential equations. Modeling physical
situations that exhibit exponential growth and exponential
decay. Plotting slope fields using the isocline method. (2 lectures / 5
pages. BB: Sections 1.1 and supplemental material. PBA: Sections 1.1,
2.1 and supplemental material.)

**Week
2**: Sketching slope fields of autonomous differential
equations. Euler's method. Solving 1st order ODEs using separation of
variables. Models of motion with air resistance. (4 lectures / 12
pages. BB: Sections 1.3, 2.1, 2.2, 2.3, 2.5. PBA: Sections 2.1, 2.2,
2.3, 2.9, 6.1)

**Week
3**: Solving 1st order linear ODEs using the integrating factor
method, and the method of variation of parameters. Strucutre of
solutions to 1st order linear ODEs, and connections to parametrized
lines. Basic mixing problems. (4 lectures / 8 pages. BB: Sections 1.4,
2.1, 2.3. PBA: Sections 2.4, 2.5.)

**Week
4**: More complicated mixing problems. The logistic equation as
a population model. Intro to 2nd order ODEs. (4 lectures / 11
pages. BB: Sections 2.3, 2.5, 4.1, 4.2, 4.3. PBA: Sections 2.5, 3.1,
4.1, 4.2, 4.3).

**Week
5**: Solving linear 2nd order ODEs. Method of undertermined
coefficients. Simple harmonic motion. (4 lectures / 10 pages. BB:
Sections 4.1, 4.2, 4.3, 4.4. PBA: Sections 4.3, 4.4, 4.5).

**Week
6**: Harmonic motion with damping and with forcing terms. (3
lectures / 7 pages. BB: Sections 4.1, 4.7. PBA: Sections 4.4, 4.7).

**Week
7**: Intro to linear algebra. Adding and multiplying
matrices. Writing systems of linear equations with matrices. Inverses
and determinants of 2x2 matrices. Eigenvalues and eigenvectors of 2x2
matrices. (4 lectures / 10 pages. BB: Sections 3.1. PBA: Sections 7.1,
7.2, 7.3, 7.4).

**Week
8**: Using linear algebra to solve systems of two 1st order
linear ODEs: x'=Ax, when then eigenvalues of A are real and
distinct. (4 lectures / 12 pages. BB: Sections 3.2, 3,3. PBA: Sections
8.1, 8.2, 8.4, 9.1).

**Week
9**: Solving x'=Ax when the eigenvalues are complex numbers. (2
lectures / 5 pages. BB: Sections 3.4. PBA: Sections 9.2, 9.3,
9.4).

**Week
10**: Solving x'=Ax when the eigenvalues are repeated. The SIR
model in epidemiology. Intro to Laplace transforms. (4 lectures / 12
pages. BB: Sections 3.5, 5.1, 5.2. PBA: Sections 5.1, 5.2, 9.2, 9.3,
9.4).

**Week
11**: More properties of Laplace transforms. Using Laplace
transforms to solve ODEs. Using the Heavyside function to express, and
take the Laplace transform of, piecewise continuous functions. (3
lectures / 7 pages. BB: 5.2, 5.3, 5.4, 5.5. PBA: 5.2, 5.3, 5.4,
5.5).

**Week
12**: Solving ODEs with discontinuous forcing terms. Taking the
Laplace transform of periodic functions. Delta functions. (4 lectures
/ 7 pages. BB: 5.5, 5.6, 5.7. PBA: 5.5, 5.6).

**Week
13**: Introduction to Fourier series -- derivation and
computation. Even and odd functions. (4 lectures / 13
pages. Supplemental material.)

**Week
14**: Complex version of Fourier series. Intro to Partial
Differential Equations (PDEs). The (1-dimensional) heat
equation. Analysis of different boundary conditions. (4 lectures / 12
pages. Supplemental material.)

**Week
15**: The (1-dimensional) wave equation. (2 lectures / 6
pages. Supplemental material.)

**Week
16**: Introduction to PDEs in higher dimensions. Harmonic
functions and Laplace's equation. Steady-state solutions to the heat
equation. Solving Laplace's equation, the heat equation, and the wave
equation in two dimensions. (3 lectures / 12 pages. Supplemental
material.)

### Homework

Homework 1. Due
Wednesday, August 25th at 4pm.

Homework 2. Due
Monday, August 31st at 4pm.

Homework 3. Due
Friday, September 4th at 4pm.

Homework 4. Due
Tuesday, September 8th at 4pm.

Homework 5. Due
Friday, September 11th at 4pm.

Homework 6. Due
Monday, September 14th at 4pm.

Homework 7. Due
Friday, September 18th at 4pm.

Homework 8. Due
Wednesday, September 23rd at 4pm.

Homework 9. Due
Monday, September 28th at 4pm.

Homework 10. Due
Monday, October 5th at 4pm.

Homework 11. Due
Wednesday, October 14th at 4pm.

Homework 12. Due
Wednesday, October 21st at 4pm.

Homework 13. Due
Wednesday, October 28th at 4pm.

Homework 14. Due
Monday, November 2nd at 4pm.

Homework 15. Due
Friday, November 6th at 4pm.

Homework 16. Due
Tuesday, November 10th at 4pm.

Homework 17. Due
Friday, November 13th at 4pm.

Homework 18. Due
Monday, November 16th at 4pm.

Homework 19. Due
Friday, November 20th at 4pm.

Homework 20. Due
Tuesday, November 24th at 4pm.

Homework 21. Due
Tuesday, December 1st at 4pm.

Homework 22. Due
Friday, December 4th at 4pm.