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### Final Exam Information

#### Schedule

This schedule will be updated as necessary throughout the semester. Section numbers refer to the text Differential Equations, J. Brannan & W. Boyce; John Wiley & Sons, 3rd edition, 2015.

Week 1:   8/24-8/28

 Mon Sec 1.1 Notes Modeling — How differential equations are used to describe real-world processes. Wed Sec 1.3 Notes Definitions, Classification & Terminology — Solutions of differential equations, the order of differential equations, linear vs. nonlinear differential equations. Fri Sec 1.2 Notes   Worksheet Group Work:   Qualitative Methods — Determining the behavior of solutions to a differential equation, without actually solving the equation.

Week 2:   8/31-9/4

 Mon Sec 2.1 Notes   Worksheet Group Work:   Separable Equations — Solution technique and interval of existence of solutions for first order separable differential equations. Wed Sec 2.2 Notes   Worksheet Linear Equations — Solving first order linear differential equations using the method of integrating factors. Fri Sec 2.3 Notes Modeling with first order differential equations.

Week 3:   9/7-9/11

 Mon No Class Wed Sec 2.4/2.5 Notes (sec 2.4) Notes (sec 2.5)   Worksheet (sec 2.5) Group Work:   Two Models of Population Dynamics Fri Sec 2.6 Notes Exact first order differential equations.

Week 4:   9/14-9/18

 Mon Catch-up/Review Wed Info/Review Key   Stats Exam 1: Chapters 1 & 2 Fri Sec A1 Notes   Homework Introduction to Matrices.

Week 5:   9/21-9/25

 Mon Sec A2 Notes   Homework Solving systems of linear algebraic equations. Wed Sec A3 Notes   Worksheet Homework Group Work: Matrix determinants and inverses. Fri Sec A4 Notes   Homework Eigenvalues and eigenvectors.

Week 6:   9/28-10/2

 Mon Sec 3.1 Notes Complex eigenvalues and eigenvectors, and a “review” of complex arithmetic. Wed Sec 3.2 Notes   Worksheet Group Work: Introduction to systems of first order linear differential equations. Fri Sec 3.3 Notes (updated 10/5 to correct errors and add information) Solving $2 \times 2$ homogenous systems of first order linear differential equations with constant coefficients. Case 1: the coefficient matrix has two distinct real eigenvalues.

Week 7:   10/5-10/9

 Mon Sec 3.3/3.4 Notes Solving $2 \times 2$ homogenous systems of first order linear differential equations with constant coefficients. Case 1 (continued): the coefficient matrix has two distinct real eigenvalues. Case 2: the coefficient matrix has complex eigenvalues Wed Sec 3.4 Worksheet. Group Work: Finding solutions to and classifying phase portraits for $2 \times 2$ systems with complex eigenvalues. Fri Sec 3.5 Notes Solving $2 \times 2$ homogenous systems of first order linear differential equations with constant coefficients. Case 3: the coefficient matrix has a repeated eigenvalue.

Week 8:   10/12-10/16

 Mon Catch-up/Review Wed Info/Review Key   Stats Exam 2: Appendix, Ch 3 Fri Sec 4.1 Notes Introduction to second order differential equations.

Week 9:   10/19-10/23

 Mon Sec 4.2, 4.3 Notes Solutions to second order linear homogenous differential equations with constant coefficients. Wed Sec 4.3 Notes   Worksheet Group Work: (1) Second order linear homogenous initial value problems, and: (2) the relationship between the solution methods using the characteristic polynomial, and using the associated system. Fri Sec 4.4 Notes Undamped harmonic motion: $ay” + cy = 0$, $a,c > 0$.

Week 10:   10/26-10/30

 Mon Sec 4.4 Notes Damped harmonic motion: $ay” + by’ + cy = 0$, $a,b,c > 0$. Wed Sec 4.5 Notes   Worksheet Group Work: Introduction to the method of undetermined coefficients — the “basic” cases. Fri Sec 4.5 The method of undetermined coefficients, continued — additional cases.

Week 11:   11/2-11/6

 Mon Sec 4.6 Notes Damped forced vibrations: $ay” + by’ + cy = g(t)$, $a,b,c > 0$, $g(t) = \alpha \cos \omega t$ or $g(t) = \beta \sin \omega t$ Wed Sec 4.6 Worksheet Group Work: Undamped forced vibrations: $y” + \omega_0^2 y = A \cos \omega t$ Fri Sec 4.7 Notes Variation of parameters.

Week 12:   11/9-11/13

 Mon Catch-up/Review Wed Info/Review Key   Stats Exam 3: Ch 4 Fri Sec 5.1 Notes Introduction to the Laplace transform.

Week 13:   11/16-11/20

 Mon Sec 5.2 Notes Properties of the laplace transform. Wed Sec 5.3 Notes   Worksheet Group Work: Inverse Laplace transform. Fri Sec 5.3 Notes Solving differential equations using the Laplace transform. Basic table of Laplace transforms.

1/23-11/27:   Fall Break   –   no class

Week 14:   11/30-12/4

 Mon Sec 5.5 Notes Discontinuous functions (the “on/off” switch). Wed Sec 5.5/5.6 Notes   Worksheet Group Work: Discontinuous functions (the “on/off” switch) and differential equations with discontinuous forcing functions. Fri Sec 5.5, 5.6, 5.7 Notes Discontinuous functions (continued), periodic functions (?), and/or unit impulse function (?).

Week 15:   12/7-12/11

 Mon Sec 5.5, 5.7 Notes Periodic and/or unit impulse functions. Wed Worksheet Review: Group Work. Fri Review

Final Exam — Monday 14 December, 12-2pm: Ch 5 & Comprehensive — Info/Review