Homework 1

D. C. Lay, S. Lay, J.J. McDonald Linear Algebra and Its Applications 

SIXTH EDITION

Week 1: Systems of Linear equations.  Row reduction and Echelon Form.  ( Section1.1, Section 1.2)

Week 2: Vector Equations.  The Matrix equation Ax=b ( Section1.3, Section 1.4)

Week 3: Solution Sets of Linear Systems.  Linear Independence.  ( Section1.5, Section 1.7)

Week 4 : Introduction to Linear Transformations.  The Matrix of Linear Transformation  (Section1.8, Section 1.9)

Week 5 :  Matrix operations. The Inverse of a Matrix  (Section 2.1, Section 2.2)

Week 5 :  Characterisation of Invertible Matrices.  ( Section 2.2, Section 2.3)

Week :  Vector Spaces and Subspaces.  (Section 4.1)

Instructor, Office and Office Hours:

 Varga Kalantarov,  Sci 162,  Thursdays at 16:15 -17:15 or by appointment.

Teaching Assistants,  Office and Office Hours:

 Ahmet Nuri Cevik,   by appointment.

 Komal Khalid,  Thursdays,                          14:30 – 16: 40 ,  SCI 130

 Onur Işık,         Thursdays ,                           16:15 -17:25,   SCI 129

 Roohul Ahmed Khalid,   Thursdays,            14:30 – 16: 40 , SCI 130

Homepage: https://mysite.ku.edu.tr/vkalantarov/

Textbook: R.A. Adams and C. Essex: Calculus ( a complete course), Pearson 10th edition (Requirred).

Topics to be covered:  Vectors and vector-functions: limit, continuity, derivative and integral of a vector-function. Functions of several variables: limit, continuity, partial derivatives, chain rule, directional derivative, extreme value problems, double and triple integrals, line integrals. Vector calculus: conservative vector fields, Green’s, Stoke’s and divergence theorems.

Homework: Suggested homework problems will be assigned regularly but not collected for grading. Students are required to solve these problems in order to gain a better understanding of the subject.

Evaluation method: Students progress will be evaluated according to their performance in, the midterm and final exams. There will be two midterm exams. The Midterm Exam 1 and 2 will be held during the weeks of November 13 and December 18, respectively. The midterm exams will cover the material not covered in the earlier exams. The final exam will be a more comprehensive exam covering all the

subjects of the course. The contribution of the midterm and final exams are as follows, midterm exams 30% each,final exam 40%.

Attendance: Students are advised to attend all the lectures and problem sessions.

Make-ups: If a student misses a midterm exam and has a valid excuse, his (her) grade in the final exam will be substituted for the grade in the missed exam.

If a student misses the final exam and has a documented valid excuse accepted by the deans office , (s)he will be given a make-up exam at noon on the the first day of the makeup period ( according to the academic calender)

Suggested method of study: Reading the lecture notes and the books is necessary for grasping the subject, but it is by no means sufficient.

Students must try to reproduce the definitions and proofs of the theorems, and apply the techniques to other problems.

Academic dishonesty: If a student is caught cheating in an exam, s/he will be punished according to the Y\”{O}K regulations.

These consist of one or two semesters of prohibition from attending the university.

Problems from Adams and Essex (10th Ed.) : 

Week 1 :

Section 10.1,  Page 574: 13, 18, 37;

 Section 10.2, Page 583: 3, 13, 20, 23;

 Section 10.3,  Page 592: 4, 15, 17, 26;

Week 2 :

Problems from the textbook.\\

{\bf Section 10.4} Page 599: 4, 8, 10, 16, 19, 22, 26

{\bf Section 12.1} Page 653: 10, 21, 24, 26

Week 3 :

Problems from the textbook.

 Section 13.1 Page 702, Problems : 4, 6, 15, 20

 Section 13.2  Page 707, Problems  4, 7, 12, 14, 20

Week 4 :

Problems from the textbook.

 Section 13.3,  Page 714, Problems : 7, 17, 19, 20, 28, 30

 Section 13.4, Page 720,  Problems : 4, 11, 13, 15

Week 5 :

Problems from the textbook.

 Section 13.5,  Page 729, Problems : 2, 6, 11, 15, 19, 22

 Section 13.6, Page 740,  Problems : 2, 5, 21, 22

Section 13.7, Page 751,  Problems : 5, 7, 12, 17, 19, 24, 31

Week  6:

Problems from the textbook.

Section 4.10 p. 283: Problems 3, 23, 34
Section 14.1 p. 777 : Problems 3, 4, 13, 22, 25
Section 14.2 p. 783: Problems 5, 9, 13

Week 1 :  Sections 10.1 -10.4: Trangle inequality.  Distance between points on plane and in space. Open and closed sets. Vectors , dot product and cross product of vectors.  Equations of plane and line in space.

Week 2 :  Sections 10.3 -10.4:  Scalar triple product: volume of a parallelepiped. Equation of plane. Equation of line. Distancew from a point to a plane. Distance from a point to a line. Intersections of planes and lines. 

Week 1    (Sections :  (1.3, 2.1, 2.2, 2.4) Basic mathematical models. Classification of differential equations. First order linear equations. Separable differential equations. Uniqueness of solution of linear first order ODE.

Week 2    (Sections :  (2.4, 2.6, 2.8)  Bernoulli equation. Exact differential equations and integrating factors. The existence and uniqueness theorem of the initial value problem for first order  ODE’s.

Week 3   (Sections :  (2.8, 3.1, 3.2)   The existence and uniqueness theorem (Picard’s iterations). Second order linear ODE’s with  constant coefficients. Existence and uniqueness of solutions to the Cauchy problem. Abel’s theorem.

Week 4   (Sections :  (3.2, 3.3, 3.4)

Applications of Abel’s theorem. Complex roots of characteristic equation. Reduction of order of second order linear ODE’s

Week 5  (Sections :  (3.6, 3.8, 5.1) Non-homogeneous second order ODE’s: method of variation of parameters. Forced periodic vibrations. Review of power series.

Week 6  (Sections :  (5.2, 5.3, 6.1) Series solutions of ODEs. Laplace transform and its properties. 

Week 7  (Sections :  (6.2, 6.3, 6.4, 6.6. )  Solution of initial value problems by using the Laplace transform. Step functions,. ODEs  with discontinuous forcing functions. The convolution integral.

Week 8  (Sections :  ( 7.1, 7.2, 7.3) Systems of of first-order linear ODE’s. Matrices . Systems of linear algebraic equations. Eigenvalues and eigenvectors of matrices.

Week 9  (Sections :  (7.4, 7.5, 7.6) Basic theory of systems of first-order linear ODE’s, Abel’s Theorem. Homogeneous linear systems with constant coefficients. Complex valued eigenvalues.

Week 10  (Sections :  (7.7, 7.8,  7.9)  Fundamental Matrices. Repeated eigenvalues, Non-homogeneous systems.

Week 11  (Sections :  (10.1, 10.2, 10.3, 10.4) ,Two-point boundary value problems. Fourier series, The Fourier convergence Theorem.  Even and odd functions (Fourier cosine and sine series).

MATH 204: DIFFERENTIAL EQUATIONS, SPRING 2023

Problems from W.E. Boyce, R.C. Diprima, D.B. Meade :

Homework 6

Section 5.1  p. 202, Problems : 2(a,b), 7 (a,b),  9(a,b)

Section 6.1   p. 245, Problems: 4, 7, 16, 20

Section 6.2  p. 253, Problems: 4, 10, 13, 20, 24

Section 6. 3  p. 253, Problems:  7, 10, 18

Homework 5

Section 3. 6 p. 144, Problems : 6, 9, 12, 16, 17

Section 3. 8 p. 165, Problems : 9, 13(a,b), 15(a)

Section 5.1  p. 193, Problems : 4, 10,   13, 17 

Homework 4 

Section 3.2 p. 117, Problems : 9, 12, 18, 27, 29

Section 3.3 p. 123, Problems : 11, 13, 16, 18 (a,b),

Section 3. 4 p. 130, Problems : 5, 15.

Homework 3.

Problems from W.E. Boyce, R.C. Diprima, D.B. Meade :

 Section 2.8 p. 89, Problems: 2, 5, 7, 11, 14 .

 Section 3.1 p. 107, Problems : 4, 5, 13, 15, 16, 20

and the following problem 

Problem A.  Show that if w(t) is continuous, non-negative on some interval [0, ∞)and satisfies the inequality

w(t)≤  ∫^t0  s² w(s)ds ,    for all t ∈ [0, ∞)

and w(0)=0, for all t ∈ [0, ∞).

Homework 1.

Problems from W.E. Boyce, R.C. Diprima, D.B. Meade :

Section 2.1   p. 31: Problems 6, 10, 24, 26 .

 Section 2.2  p. 38 : Problems 6, 9, 17, 20, 23.