MTHSC 208, Fall 2008

# MTHSC 208, Fall 2008

"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 Monday, August 25th 2008 at 4pm.

Homework 2: Section 2.1: #31. Section 2.2: #16-20, 35. Section 2.3: #2, 3, 10, 12, 18, 19. Due Friday, August 29th at 12:20 pm. For #31, use the isoclines method to plot the curves (not in the book; to be done in class on Wednesday). You don't need it, but the general solution is y(t)=3/(1+Ce^(-3t)). Write a sentence or two of how this relates to population dynamics; e.g., what it might be a model for and why.

Homework 3: Section 2.3: #8. Section 2.4: #1-6, 14-16, 18-21, 29. Additionally, use the isocline method to sketch the slope fields for #1, 2, and 4. Due Monday, September 1st 2008 at 4pm.

Homework 4: Section 2.4: #36-44. Section 2.5: #1-4. Due Friday, September 5th 2008 at 4pm.

Homework 5: Due Tuesday, September 9th 2008 at 4pm.

Homework 6: Section 3.3: #1, 2. Section 6.1: #1, 2, 6. Section 4.1: #1-8. Due Friday, September 12th 2008 at 4pm.

Homework 7: Section 4.1: #10-12, 13, 16-18, 22, 23, 26, 28, 30. Due Tuesday, September 16th 2008, at 4pm.

Homework 8: Section 4.3: #1-4, 9-14, 17-20, 25-28, 37, 38. Section 4.2: 1-4, 8. Due Friday, September 19th 2008, at 4pm.

Homework 9: Section 2.2: #27, 34. Section 2.3: #4. Section 2.4: #8, 17, 33, 34 (Also, sketch the slope field using isoclines for 33 & 34). Section 2.5: #5. Section 2.9: #18, 30, 31 (For 30 & 31, "qualitative analysis" means "don't actually do the math"!). Section 3.1: #10. Section 4.1: #24, 27. Section 4.3: #32-34. Section 4.4: #1, 2, 7, 8. Section 6.1: #3, 4. Due Wednesday, September 24th, at 4pm.

Homework 10: Section 4.5: #9, 18, 20, 22-24, 38, 40. Section 4.7: #2, 4, 6, 7. Due Tuesday, September 30th, 2008 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, October 3 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 Monday October 6th at 4pm.

Homework 13: Section 5.4 #4, 20. Section 5.5 #2, 4-6, 9, 13, 14, 16, 18, 20. Due Friday October 10th at 4pm.

Homework 14: Due Friday, October 17th 2008 at 4pm.

Homework 15: Due Tuesday, October 21st 2008 at 4pm.

Homework 16: Due Friday, October 24th 2008 at 4pm.

Homework 17: Due Friday, October 31th 2008 at 4pm.

Homework 18: Due Wednesday, November 5th 2008 at 4pm.

Homework 19: Due Friday November 7th 2008 at 4pm.

Homework 20: Due Wednesday November 12th 2008 at 4pm.

Homework 21: Due Friday November 14th 2008 at 4pm.

Homework 22: Due Thursday November 20th 2008, in class.

Homework 23: Due Friday, December 5th 2008, in class.

### Main topics covered in class

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 variation of parameters. Plotting direction fields using isoclines. (Sections 2.3, 2.4 & supplemental material).

Week 3: Solving linear equations by variation of parameters. Mixing problems. Autonomous equations. (Sections 2.4, 2.5, 2.9).

Week 4: Euler's method. Intro to 2nd order ODEs. (Sections 4.1, 6.1).

Week 5: Solving linear homogeneous 2nd order ODEs with constant coefficients. Systems of 1st order ODEs. Harmonic motion. (Sections 4.2, 4,3, 4.4)

Week 6: Solving linear inhomogeneous 2nd order ODEs with constant coefficients, using the method of undetermined coefficients. Harmonic motion with a forcing term (but no damping). (Sections 4.5, 4.7)

Week 7: Laplace and inverse Laplace transforms. (Sections 5.1, 5.2, 5.3, 5.4)

Week 8: Laplace and inverse Laplace transforms. (Sections 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: More ODEs that have power series solutions. Basic linear algebra -- vector spaces, bases, and connections to homogeneous ODEs. Ordinary and singular points of ODEs. Generalized power series and the method of Frobenius (Sections 11.3, 11.4, 11.5, and supplemental material).

Week 11: Inner products on vector spaces. Fourier series -- derivation and computation (Sections 12.1 and supplemental material). Week 11 class notes.

Week 12: More on Fourier series. Even and odd functions. Fourier sine and cosine series. The complex form of Fourier series. (Sections 12.1, 12.3, 12.4). Week 12 class notes.

Week 13: Intro to PDEs. The heat equation. Solving the heat equation by separation of variables, and using Fourier series to solve the initial value problem. The wave equation (introduced). Week 13 class notes.

Week 14: Solving the wave equation. Analysis of boundary and initial conditions of PDEs. Introduction to higher-dimensional PDEs. Harmonic functions and Laplace's equation. Solving the 2D Laplace equation, and the 2D heat equation. Week 14 class notes.

Week 15: Solving the 2D 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. Week 15 class notes.