## MATH 3740: Differential Equations and Linear Algebra

.   Schedule for Week 8 .   Quiz on Tuesday,  problems from Section 8.1.  Final exam (comprehensive) on Friday, August 14.

### Calendar

#### Course description: This course includes most of the standard material for differential equations: first-order differential equations and analytic methods of their solution, mathematical modeling using differential equations, numerical methods for solving differential equations, linear equations of higher order, systems of linear and nonlinear equations, stability of equilibrium solutions, Laplace transform methods. In addition some related material from linear algebra is included. We use a computer algebra system Maple  in this course for some computer projects.  It is expected that you will work most of the basic exercises at the end of each section. A list of additional problems to work will be placed at the course webpage (see homework problems). It is recommended to  write down solutions of these problems and keep your homework to use in  preparation for quizzes, mid-term and final exams. Every week we'll have a quiz containing two or three problems analogous to ones solved in class and from homework. If you have any questions about problems, please ask them in class or  during office hours.

Textbook:  C.H.Edwards,Jr. and D.E.Penney , Differential Equations and Linear Algebra , (Third and Fourth Editions) Prentice Hall.
Complementary useful reading for problems and solutions manual is available on the Web via WMU Library page in the electronic form: Bronson, R.     Schaum’s outline of theory and problems of differential equations [electronic resource]

Office Hours:  Monday,Tuesday, Thursday and Friday: 11:10 am  - 12:00 pm.

### Week 8:  August 10 - August 14

• Section 9.1 Phase portraits. Stability and asymptotic stability of solutions of linear systems. Equilibrium solutions.
• Section 9.2 Nonlinear Systems - their critical points and equilibrium solutions. Stability of equilibrium solutions. Linearization of systems near equilibrium solutions.
• Section 10.1 Laplace transforms and inverse transforms
• Review of the course
Quiz on Tuesday,  problems from Section 8.1.  Final exam (comprehensive) on Friday, August 14.

### Week 7:  August 3 - August 7

Topics for study and reading :
• Section 7.3 Eigenvalue method for linear systems ( including complex eigenvalues).
• Section 7.5 Multiple eigenvalue solutions.
• Section 8.1 Matrix exponential and linear systems : fundamental matrix solution and exponential matrix.
• Section 8.2 Nonhomogeneous linear system and method of variation of parameters. Cauchy's  formula.
No quizzes  on Tuesday and Friday.
Second midterm exam on Thursday, August 6,
will include  topics  from  Chapter 5 (Sections 5.1-5.6) and Chapter 7 (Sections 7.1-7.3):

### Week 6:July 27 - July 31

Topics for study and reading :
• Section 5.4 Mechanical vibrations: simple pendulum, spring-dashpot mechanical system. Free  undamped motion: amplitude, circular frequency and phase angle. Free damped motion.
• Section  5.5 Non-homogeneous equations and the method of undetermined coefficients.
• Section 5.6 Forced oscillations and resonance
• Section 7.1 First-order systems of differential equations. Transformation of higher-order differential equation   to first-order system. Simple two-dimensional systems.
• Section 7.2.Matrices and linear systems.
• Section 7.3 Eigenvalue method for linear systems ( including complex eigenvalues).
Quiz on Tuesday, July 28: problems from section 5.3, 5.4. Quiz on Friday : problems from Sections 5.5,5.6, 7.1 and 7.2.

### Week 5: July 20 - July 24

Topics for study and reading :
• Section 4.3: Linear combinations, linear dependence and independence of vectors
• Section 5.1   Linear     second-order differential equations   and their general solutions
• Sections 5.2 Linear n-th order differential equations.Existence and  uniqueness of solutions. Superposition principle. Linear dependence of   solutions, wronskians.
General solutions  homogeneous linear differential equations.
• Section 5.3 Linear n-th  order linear differential equations with constant coefficients. Characteristic equation. Cases of  distinct and repeated roots. Complex numbers, Euler formula..
• Section 5.4 Mechanical vibrations: simple pendulum, spring-dashpot mechanical system. Free  undamped motion: amplitude, circular frequency and phase angle. Free damped motion.
Quiz on Tuesday, July 21: problems from section 3.3,3.4,3.5. Quiz on Friday: problems from Sections 5.1, 5.2.

### Week 4:July 13- July 17

• Sections 2.4, 25   Numerical approximation of solutions: Euler's method, improved Euler method,    Runge-Kutta method.
• Sections 3.1-3.3:  Linear system of algebraic equations and their general solutions. Gauss-Jordan elimination method.
• Chapter 3:  Elements of linear algebra: vector and matrix operations, matrix product, determinants, inverse matrices.
• Chapter 3: Determinants, inverse matrices, homogeneous linear systems of equations.
• Section 4.3: Linear combinations, linear dependence and independence of vectors
.

First mid-term exam  on FRIDAY. July 17   (sample exam) will cover Chapters 1 and 2 (sections 2.1-2.3) and Chapter 3 (Sections 3.1-3.3).
Maple computer project #2 is due to Monday, July 20.

### Week 3: July 6 - July 10

• Section 2.2 Equilibrium solutions and their stability.
• Section 2.3 Acceleration-velocity models. Motion under resistance forces.
• Computer algebra system Maple:  Introduction ( basic capabilities, plotting, programming, Differential Equations packages)
• Sections 2.4, 25   Numerical approximation of solutions: Euler's method, improved Euler method,    Runge-Kutta method.

Quiz on Tuesday: problems from Sections 2.1, 2.2. Quiz on Friday: problems from Section 2.3. Maple Computer Project #1 is due to  Monday, July 13.

### Week 2:June 29  - July 03

• Section 1.4 Separable equations and their solution by separation of variables. Implicit solutions.
• Section 1.5 Linear first-order equations: uniqueness and existence of solutions, integrating factor and    solutions of linear first-order  differential equation.
• Section 1.6 Substitution methods and exact solutions. Substitution (or, change of variables) method,   homogeneous equations, Bernoulli equations. Exact differential equations and criterion for exactness.
• Section 2.1  Mathematical models: population models. Logistic equation and its applications.
• Section 2.2 Equilibrium solutions and their stability.
•  Quiz on Tuesday: problems from Sections 1.3,1.4. Quiz on Friday:  problems from Sections  1.4, 1.5, 1.6, theorem on existence and uniqueness of solutions

### Week 1: June 25 - January 26

#### Topics for study and reading

• Section 1.1 Differential equations and mathematical models: Newton's law of cooling. Concept of solution of differential equation.
Initial conditions and initial value problems.
• Section 1.2 Integrals as general and particular solutions. Examples - motion problems.
• Section 1.3 Directional fields. Existence and uniqueness of solutions of initial value problem for differential equations.
• Section 1.4 Separable equations and their solution by separation of variables. Implicit solutions.

Quiz on Friday:  problems from Sections 1.1, 1.2 , similar to   homework problems   for 4th edition, homework problems for 3rd edition.

## Homework Problems for 4th edition

Note that problems marked  with *  are extra.

#### Section 1.6 (page 70)  1, 2, 8, 11, 14, 15, 16, 19, 20, 21, 32, 34.

Section  2.1 (page 82) 1, 3, 5, 7, 8, 11, 16, 27, 28

Section 2.2  (page 92)   1, 3, 6, 9, 11, 14, 15, 21

Section 2.3  (page 101)   2, 4, 7, 10, 11, 20, 28

Section 2.4 (page 114)  1, 3, 4, 9, 12, 14

Section 2.6 (page 133)  2, 4, 14, 15

Section 3.1 (page 145) 1, 5, 7, 11, 17, 23, 31

Section 3.2 (page 154) 1, 5, 11, 19, 24*,

Section 3.3 (page 162) 1, 3, 17

Section 3.4 (page 173) 1, 3, 5, 7, 14, 17, 27*

Section 3.5 (page 185) 1, 3, 25, 30, 32

Section 3.6 (page 201) 1, 2, 47, 48, 52

Section 4.3 (page 234) 17, 18, 19

Section 5.1 (page 276)   1, 5, 8, 11, 12, 13, 14, 16, 20, 23, 26, 27, 33, 34, 36. , 41, 43, 46, 48,

Section 5.2 (page 288) 1,  4, 7, 9, 11, 13, 14, 18, 21, 24

Section 5.3 (page 300)  1,4,5,6, 11,13, 20, 21, 24, 25, 27, 30, 33, 39, 43

Section 5.4 (page 311)  1, 3, 4, 5, 8, , 10, 13, 15, 24, 27

Section 5.5 (page 325) 1, 4, 7, 11, 12, 17, 21, 26, 36, 41, 44,  47, 58

Section 5.6 (page 335) 2, 5, 10, 15, 20, 22, 26, 28, 30

Section 7.1  (page 371)  1, 2, 5, 8, , 9, 11, 15, 21, 28

Section 7.2 (page 384)  1, 3 , 5, 15, 24, 28, 37

Section  7.3  (page 395) 1, 3, 7, 9, 17, 19, 25, 38

Section 7.6 (page 451)  1, 3, 6, 7

Section  8.1 (page 479) 1, 3, 5, 9, 11, 13

Section 8.2 (page 489) 1, 3, 5, 7

Section 9.1 (page 511) 9, 11, 13, 14, 15, 20

Section 9.2 (page 522) 1,3, 5, 7. 9. 19. 21. 27. 29

## Homework Problems for 3rd edition

Note that problems marked  with *  are extra.

#### Section 1.6 (page 74)  1, 2, 8, 11, 14, 15, 16, 19, 20, 21, 32, 34.

Section  2.1 (page 87) 1, 3, 5, 7, 8, 11, 16, 27, 28

Section 2.2  (page 96)   1, 3, 6, 9, 11, 14, 15, 21

Section 2.3  (page 106)   2, 4, 7, 10, 11, 20, 28

Section 2.4 (page 119)  1, 3, 4, 9, 12, 14

Section 2.6 (page 141)  2, 4, 14, 15

Section 3.1 (page 152) 1, 5, 7, 11, 17, 23, 31

Section 3.2 (page 162) 1, 5, 11, 19, 24*,

Section 3.3 (page 170) 1, 3, 17

Section 3.4 (page 182) 1, 3, 5, 7, 14, 17, 27*

Section 3.5 (page 194) 1, 3, 25, 30, 32

Section 3.6 (page 212) 1, 2, 47, 48, 52

Section 4.3 (page 248) 17, 18, 19

Section 5.1 (page 294)   1, 5, 8, 11, 12, 13, 14, 16, 20, 23, 26, 27, 33, 34, 36. , 41, 43, 46, 48,

Section 5.2 (page 306) 1,  4, 7, 9, 11, 13, 14, 18, 21, 24

Section 5.3 (page 319)  1,4,5,6, 11,13, 20, 21, 24, 25, 27, 30, 33, 39, 43,

Section 5.4 (page 331)  1, 3, 4, 5, 8, , 10, 13, 15, 24, 27

Section 5.5 (page 346) 1, 4, 7, 11, 12, 17, 21, 26, 36, 41, 44,  47, 58

Section 5.6 (page 357) 2, 5, 10, 15, 20, 22, 26, 28, 30

Section 7.1  (page 400)  1, 2, 5, 8, , 9, 11, 15, 21, 28

Section 7.2 (page 413)  1, 3 , 5, 15, 24, 28, 37

Section  7.3  (page 426) 1, 3, 7, 9, 17, 19, 25, 38

Section 7.5  (page 457)  1, 3, 6, 7

Section  8.1 (page 487) 1, 3, 5, 9, 11, 13

Section 8.2 (page 496) 1, 3, 5, 7

Section 9.1 (page 519) 9, 11, 13, 14, 15, 20

Section 9.2 (page 532) 1,3, 5, 7. 9. 19. 21. 27. 29

Computer Project #2 :  You can work in 2-persons teams for this project and submit your work  as a team. Please,

## Exams

• First mid-term exam :  Thursday, July 16 . Chapters 1, 2 (Sect.2.1-2.3), Chapter 3 (sections 3.1-3.3) (sample exam)
• Second mid-term exam:  Thursday,  August 6   (sample exam)
• Final exam : Friday, August 14, 1:00 p.m. to 3:00 p.m.

Grades will be assigned using a sliding scale. The usual passing level "C" is between 65% and 68\%. Someone with 88\% may get "A". After each test we'll get an approximate passing level. Do not panic if your grade seems low, come and talk with me  about your performance.Various numbers below  determine  the relative weights of exams' and quizzes'  grades: Total number of points 500. The final exam - 200 points. Each midterm exam - 100 points,  quizzes grade - 100.

In general, there will be NO makeups for exams. If you miss an exam for a valid well-documented reason then arrangements may be made on an individual basis.

Students are responsible for making themselves aware of and understanding the University policies and procedures that pertain to Academic Honesty. These policies include cheating, fabrication, falsification and forgery, multiple submission, plagiarism, complicity and computer misuse. The academic policies addressing Student Rights and Responsibilities can be found in the Undergraduate Catalog at http://catalog.wmich.edu/content.php?catoid=24&navoid=974 and the Graduate Catalog at http://catalog.wmich.edu/content.php?catoid=25&navoid=1030. If there is reason to believe you have been involved in academic dishonesty, you will be referred to the Office of Student Conduct. You will be given the opportunity to review the charge(s) and if you believe you are not responsible, you will have the opportunity for a hearing. You should consult with your instructor if you are uncertain about an issue of academic honesty prior to the submission of an assignment or test.

Students and instructors are responsible for making themselves aware of and abiding by the “Western Michigan University Sexual and Gender-Based Harassment and Violence, Intimate Partner Violence, and Stalking Policy and Procedures” related to prohibited sexual misconduct under Title IX, the Clery Act and the Violence Against Women Act (VAWA) and Campus Safe. Under this policy, responsible employees (including instructors) are required to report claims of sexual misconduct to the Title IX Coordinator or designee (located in the Office of Institutional Equity). Responsible employees are not confidential resources. For a complete list of resources and more information about the policy see www.wmich.edu/sexualmisconduct.

In addition, students are encouraged to access the Code of Conduct, as well as resources and general academic policies on such issues as diversity, religious observance, and student disabilities:
·        Office of Student Conduct www.wmich.edu/conduct
·        Division of Student Affairs www.wmich.edu/students/diversity
·        University Relations Office http://www.wmich.edu/policies/religious-observances-policy
·        Disability Services for Students www.wmich.edu/disabilityservices

Comments, questions, problems to  ledyaev@wmich.edu .