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

### Class lecture notes (About these notes)

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.