M 427K. Advanced Calculus for Applications I-Engineering Honors.
Unique #54285, QR Flag, Spring 2015


Prof. Todd Arbogast
E-Mail: arbogast@ices.utexas.edu
Office: RLM 11.162, Phone: 512-471-0166
Office hours: MTWTh 8:30-9:30 a.m.

Teaching Assistant

Mr. Roman Fayvisovich
E-mail: rfayvisovich@math.utexas.edu
Office: RLM 11.152, Phone: 512-471-1684
Office hours: T 2:00-4:00 p.m.


M 408D, M 408L, or M 408S with a grade of at least C-.


TTh 11:00-12:30 p.m. in CPE 2.214 (lecture) and MW 11:00-12:00 noon in BEL 328 (discussion). Attendance is required at all class meetings.


Differential Equations and their Applications, 4th ed., by M. Braun, Springer, 1993.

Class Web Site

We will use the University's CLIPS and Canvas web sites. Please check that your scores are recorded correctly in Canvas.

Course Outline

We plan to cover the following textbook chapters and sections.

Chapter 1. First-order differential equations (supplement with direction fields) [4 hours]
   1.1 Introduction
   1.2 First-order linear differential equations
   1.4 Separable equations
   1.9 Exact equations, and why we cannot solve very many differential equations
   1.10 The existence-uniqueness theorem; Picard iteration

Chapter 2. Second-order linear differential equations [12 hours]
   2.1 Algebraic properties of solutions
   2.2 Linear equations with constant coefficients
      2.2.1 Complex roots
      2.2.2 Equal roots; reduction of order
   2.3 The nonhomogeneous equation
   2.4 The method of variation of parameters
   2.5 The method of judicious guessing
   2.6 Mechanical vibrations
   2.8 Series solutions
      2.8.1 Singular points, Euler equations
      2.8.2 Regular singular points, the method of Frobenius

Chapter 3. Systems of differential equations (with supplementary material added) [14 hours]
   3.1 Algebraic properties of solutions of linear systems
      3.A Matrix multiplication as linear combinations of columns
   3.2 Vector spaces
      3.B Vectors as arrows in Rn and a geometric meaning of operations
      3.C Complete solution system of linear equations (RREF)
      3.D Null and column spaces of a matrix
   3.3 Dimension of a vector space
   3.4 Applications of linear algebra to differential equations
   3.5 The theory of determinants
   3.6 Solutions of simultaneous linear equations
   3.7 Linear transformations
   3.8 The eigenvalue-eigenvector method of finding solutions
   3.9 Complex roots
   3.10 Equal roots
   3.11 Fundamental matrix solutions; eAt

Chapter 4. Qualitative theory of differential equations [3 hours]
   4.1 Introduction
   4.2 Stability of linear systems
   4.4 The phase-plane
   4.7 Phase portraits of linear systems

Chapter 5. Separation of variables and Fourier series [8 hours]
   5.1 Two point boundary-value problems
   5.2 Introduction to partial differential equations
   5.3 The heat equation; separation of variables
   5.4 Fourier series
   5.5 Even and odd functions
   5.6 Return to the heat equation

Chapter 6. Sturm-Liouville boundary value problems [optional 3 hours, if time permits]
   6.1 Introduction
   6.2 Inner product spaces
   6.3 Orthogonal bases, Hermitian operators
   6.4 Sturm-Liouville theory

Computer Accounts

A computer account on the Mathematics Department network can be obtained in the Undergraduate Computer Lab, RLM 7.122.

Homework and Quizzes

Homework will be assigned weekly, with only a portion to be fully graded, and due generally on Wednesdays. It is acceptable for groups of students to help each other with the homework exercises; however, each student must write up his or her own work. Late homework will not be accepted for credit (unless there is a valid health issue), and homework must be turned in during class. The textbook has answers to the odd numbered exercises. Quizzes will be given periodically on Mondays or Wednesdays in the discussion sessions.


Three exams will be given on Wednesdays, February 11, March 11, and April 15. A comprehensive final exam will be given during finals week on Friday, May 15, 2:00-5:00 p.m.

Final Grade

Grades on the three midterm exams will be scaled to count 20 points each. For the homework and quizzes, the two lowest scores will be dropped, and the result will count as 20 points. The final exam will count 40 points. The final grade on the letter plus/minus scale will be determined out of 100 points by dropping the lowest midterm test grade, or by weighting the final test grade by 1/2 (i.e., count it as 20 points). The homework and quiz score will count in the final grade.

Student Honor Code

"As a student of The University of Texas at Austin, I shall abide by the core values of the University and uphold academic integrity."

Code of Conduct

The core values of The University of Texas at Austin are learning, discovery, freedom, leadership, individual opportunity, and responsibility. Each member of the university is expected to uphold these values through integrity, honesty, trust, fairness, and respect toward peers and community.

Students with Disabilities

The University provides upon request appropriate academic accommodations for qualified students with disabilities. Contact the Office of the Dean of Students at 512-471-6259, 512-471-4641 TTY, and notify your instructor early in the semester.

Religious Holidays

Appropriate academic accommodation for major religious holidays is provided upon request.

Emergency Classroom Evacuation

Occupants of University of Texas buildings are required to evacuate when a fire alarm is activated. Alarm activation or announcement requires exiting and assembling outside. Familiarize yourself with all exit doors of each classroom and building you may occupy. Remember that the nearest exit door may not be the one you used when entering the building. Do not re-enter a building unless given instructions by the Austin Fire Department, the University Police Department, or the Fire Prevention Services office.