Class 1 (6/29/09): We covered the background and purposes of differential equations, and how to draw slope fields and phase lines.
Class 2 (6/30/09): The role of parameters, bifurcations, and Euler's method.
Homework (not collected):
Problems 1.3.7,11,15
1.4.1,3,6,13,20
1.6.1,7,9,13,21,32,33,39
1.7.3,5,9,13,17
Class 3 (7/2/09): We thought about existence and uniqueness, and learned the properties of solutions to linear ODEs, and analytic techniques that work for first-order linear and first-order separable ODEs
Homework (not collected):
Problems 1.2.1,3,19,21
1.9.1,5,6,11
Class 4 (7/6/09): We wrapped up Chapter 1 with a Venn diagram and numerous examples. We then introduced higher-order ODEs, including how to make them into systems of first-order ODEs, how to visualize autonomous systems using the phase plane,check solutions, decoupling, and the harmonic oscillator.
Homework (not collected):
Chapter 1 Review Problems 3,5,10,17-20,27,29,49
2.1.1-6,19
2.2.1-6,7,13,15
Class 5 (7/7/09): We continued our discussion of general first-order systems, introducing the predator-prey model (a nonlinear first-order system), and learned how to find equilibrium solutions. We also learned Euler's method for systems, and applied this method to the harmonic oscillator model.
Homework (not collected):
2.4.1,5
Chapter 2 Review Problems 13,18-25,27,28,29
Class 6 (7/9/09): We began analyzing linear autonomous systems (Chapter 3), first using the computer, then talked about the linearity principle for such systems, and did some review of linear algebra. We also learned how to find straight-line solutions by finding the eigenvalues of a matrix. Reminder: Chapter 1 Projects due in one week!
Homework (not collected):
3.1.5,9,15,17,24,25
3.2.1,5,13,19
Class 7 (7/13/09): We finished analyzing all possible linear, autonomous, first-order systems by discussing those with complex, repeated, and zero eigenvalues, and organized it all using the trace-determinant plane
Class 8 (7/14/09): We finished our discussion from class 7, and reviewed for the exam
Homework (not collected):
3.3.17,19,21-27
3.4.1,3,8,9,13,21
3.5.1-8,12,15,16,19
Review Problems 19,21,23-32
Class 9 (7/16/09): Midtem Exam
Class 10 (7/20/09): We covered Section 4.1 (linear NONhomogeneous second-order ODEs),reviewing the linearity priciple and its application, and practicing the method of Undetermined coefficients (guessing). We also started discussing Resonance.
Homework (not collected):
4.1.1,5,7,19,25,33,39,41
Class 11 (7/21/09): We went over the exam and chose labs. That's it.
Class 12 (7/23/09): We covered Section 4.2 and 4.3, focusing on finding particular solutions for many kinds of sinusoidal forcing functions, qualitatively analyzed the beating, and analytically solved the resonant case for the undamped harmonic oscillator.
Homework (not collected):
4.2.1-13,17,20
4.3.1-13,15,20,21,23
4.4.3,6
Class 13 (7/27/09): We wrapped up Chapter 4, then introduced the Laplace operator, proved some basic properties and saw how it can be used to solve ODEs by turning them into algebraic equations.
Class 14 (7/28/09): We began an intensive study of the tools used to find the Laplace and inverse Laplace transforms of many functions, and then started using the technique for second-order ODEs.
Homework (not collected):
6.1.1,3,11,15,16,19,26,27
6.2.1,3,5,9,11,13,17,19
6.3.1,3,5,7,27,29
Class 15 (7/30/09): We studied the Dirac delta function, and introduced the convolution of two functions, which is involved in the product rule for the Laplace transform.
Homework (not collected):
6.4.3,5,7,9
6.5.7,9,11
Class 16 (8/3/09): We studied section 6.6: how the poles of the the Laplace transform of the solution determine its long-term behavior, generalizing what we already knew about the roots of the characteristic equation for the unforced harmonic oscillator.
Homework (not collected):
6.6.1,3,5,6
Class 17 (8/4/09): Labs collected and review for final - Bring your questions!
Class 18 (8/6/09): Final exam - Good luck everyone!!