MTHSC 208, Spring 2009

# MTHSC 208, Spring 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

### Homework

Homework 1: Section 2.1: #6, 7, 17, 20. Section 2.2: #1-3, 13-15, 25, 26, 33, 34. [scanned copy]. Due Wednesday, January 14th 2009 at 4pm.

Homework 2: Due Tuesday, January 20th at 4:00 pm. Section 2.1: #31. Section 2.2: #16-20, 35. Section 2.3: #3, 4, 8, 10, 18, 19. For #31, use the isoclines method to plot the curves (not in the book; see in-class notes).

Homework 3: Section 2.4: #1, 2, 4, 6, 14-16, 18-21, 29. Additionally, use the isocline method to sketch the slope fields for #1, 2, and 4. Due Friday, January 23rd at 4pm.

Homework 4: Section 2.4: #36-44. Section 2.5: #1-4. Due Tuesday, January 27th at 4pm.

Homework 5. Due Friday, January 30th at 4pm.

Homework 6: Section 3.3: #1, 2. Section 6.1: #1, 2, 6, 7. Section 4.1: #1-8. For #6-7 in 6.1, use ONLY step-size h=0.2. Due Monday, February 2nd at 4pm.

Homework 7: Section 2.2: #27, 34. Section 2.3: #4. Section 2.4: #17, 33, 34 (Also, sketch the slope field using isoclines for 33 & 34). Section 2.5: #5. Section 2.9: #18, 31 (For 31, ``qualitative analysis'' means ``don't actually do the math!''). Section 3.1: #10. Section 6.1: #3. Due Thursday, February 5th at 4pm.

Homework 8: Section 4.1 #17-18, 26, 28, 30. Section 4.2 #1-4. Section 4.3 #1, 2, 9, 10, 17, 18. Due Monday, February 9th at 4pm.

Homework 9: Section 4.3 #25, 26, 29, 30, 33, 34. Section 4.4: #1, 2, 7, 8. Section 4.5: #9, 18, 22, 23. Due Friday February 13th at 4pm.

Homework 10: Section 4.5: #20, 24, 31, 32, 38, 39, 41, 44. Section 4.7: #2, 4, 6, 7. Due Monday, February 16th at 4pm.

Homework 11: Section 5.1. #2, 4, 6, 8, 10 (compute the integrals on these)
#15, 17, 19, 21, 23 (use the table to compute these)
#26, 28, 29. Due Friday, February 20th at 4pm.

Homework 12: Section 5.2 #22, 24, 34, 36, 38, 40. Section 5.3 #2-5, 8, 16, 22, 36. Section 5.4 #10, 12. Due Tuesday February 24th at 4pm.

Homework 13: Section 5.4 #2, 6, 28. Section 5.5 #1-3, 10-12. Due Friday 27th at 4pm.

Homework 14: Section 5.4 #20. Section 5.5 #4-6, 9, 13, 16, 18, 20. Section 5.6 #4, 6, 9-11. Due Monday, March 2nd at 4pm.

Homework 15. Due Wednesday, March 11th at 4pm.

Homework 16. Due Monday, March 23rd at 4pm.

Homework 17. Due Friday, March 27th at 4pm.

Homework 18. Due Monday, March 30th at 4pm.

Homework 19. Due Friday, April 3rd at 4pm.

Homework 20. Due Monday, April 6th at 4pm.

Homework 21. Due Friday, April 10th at 4pm.

Homework 22. Due Wednesday, April 15th at 4pm.

Homework 23. Due Monday, April 20th at 4pm.

Homework 24. Due Friday, April 24th at 4pm.

### Class lecture notes

Week 1: Exponential growth and decay problems. Heating a cooling problems. Solving 1st order ODEs by separation of varibles. (Sections 2.1, 2.2, 3.3).

Week 2: Falling objects, with & without air resistance. Solving linear equations by integrating factor. Plotting direction fields using isoclines. (Sections 2.3, 2.4 & supplemental material).

Week 3: Solving linear equations by variation of parameters. Connections between solutions of 1st order linear ODEs and parametrized lines (homogeneous and particular solutions). Mixing problems. Plotting autonomous ODEs. (Sections 2.3, 2.4, 2.9, & supplemenetal material).

Week 4: Logistic equation. Autonomous equations. Euler's method. Intro to 2nd order ODEs. (Sections 2.9, 3.1, 4.1, 6.1).

Week 5: Solving basic 2nd order ODEs (homogeneous, constant coefficients). Euler's equation. Writing higher-order ODEs as a system of 1st order ODEs. [Midterm 1] (Sections 4.1, 4.2, 4.3, 4.4).

Week 6: Harmonic motion. Solving 2nd order linear inhomogeneous ODEs using the method of undetermined coefficients. Forced harmonic motion. (Sections 4.4, 4.5, 4.7).

Week 7: Laplace and inverse Laplace transforms. Using them to solve ODEs. (Sections 5.1, 5.2, 5.3, 5.4).

Week 8: Using the Heavyside function to express piecewise continuous functions, and solving ODEs with discontinuous forcing terms. Delta functions. Summary of Chapter 9 -- systems of linear ODEs. (Sections 5.4, 5.5, 5.6).

Week 9: Cauchy-Euler equations. Basic power series. Radius of convergence. Solving ODEs using power series. (Sections 11.1, 11.2, supplemental material).

Week 10: Power series. Radius of convergence. Ratio test and comparison test for convergence. [Midterm 2]

Week 11: Ordinary and singular points of ODEs. Generalized power series and the method of Frobenius. Basic linear algebra -- vector spaces, bases, and connections to homogeneous ODEs. (Sections 11.3, 11.4, 11.5, and supplemental material).

Week 12: Inner products on vector spaces. Fourier series -- derivation and computation (Sections 12.1, 12.3, 12.4, and supplemental material).

Week 13: Complex Fourier series. Introduction to Partial Differential equations (PDEs). Solving the heat equation. (Sections 12.4, 13.1, 13.2).

Week 14: Analysis of boundary and initial conditions for the heat equation. The wave equation. Introduction to PDEs in higher dimensions. Harmonic functions and Laplace's equation. (Sections 13.2, 13.3, 13.4).

Week 15: Solving the 2D Laplace's equation, heat equation, and wave equation. Special topics: (1) How Laplace transforms can be used to solve PDEs, (2) Fourier transforms - an extension of Fourier series, and how they can be used to solve PDEs, (3) Wavelets - a sometimes better alternative to Fourier series. (Sections 13.4, 13.9).