
• Credits:  3.00 
 • Schedule:  Wednesday, 6:40–9:10pm 
 • Classroom:  McCortWard 114 
 • Number:  01LEC (3647) 
 • Capacity:  35 
  
 • Instructor:  Dr. René Doursat 
 • Email:  doursatcua.edu 
 • Phone:  (202) 3195160 
 • Office:  Pangborn 121 
 • Hours:  Tuesday & Thursday, 2:30–3:30pm 
  Wednesday, 5:30–6:30pm 
The objective of this course is to provide an introduction to the mathematical
methods that will be needed in subsequent graduatelevel courses in engineering.
Emphasis is placed on understanding the concepts for solving first and secondorder
differential equations (ODEs), as opposed to extricating an answer from math
packages such as Mathematica or MATLAB. The philosophy of understanding the
underlying concepts of the math is emphasized throughout the course and is
extended to the concept of integration in the complex plane and the origination
of the Fourier/Laplace transforms as well as their inverse transforms.
Additionally, various special functions are examined including: Gamma, Beta,
Bessel, and Legendre (and associated) polynomials. The process of separation
of variables in partial differential equations (PDEs) is covered in the three
major coordinate systems with application to the Laplace, heat, and wave equations.
This is a tentative list of topics, subject to modification
and reorganization.
 Firstorder ODEs
 Secondorder ODEs, Wronskian
 Power series solutions, Bessel functions, Legendre polynomials
 Convergence, regular and essential singularities
 Method of Frobenius, associated Legendre equation
 Integration techniques, complex Z and Wplanes,
multivalued functions and branch cuts
 CauchyRiemann equations, deformation of contours,
integration about a singularity
 Taylor and Laurent expansions of f(z), Jordan's lemma,
Gamma and Beta functions
 Fourier series solutions for periodic functions,
Fourier and inverse Fourier transforms
 Laplace and inverse Laplace transforms
 Classification of PDEs
 Separation of variables in Cartesian, cylindrical, and spherical
coordinates
 Solution of model equations: Laplace's, wave and heat equations
 Extended discussion of Bessel functions to include development
of second solutions, asymptotic representations, Hankel functions
(cylindrical and spherical)
 The use of laptops, tablets, cell phones, cameras,
recorders, or other devices is not allowed during any of the meetings
(lectures and exams). Bring paper and pens!
 Homework will be assigned after every class
(i.e. weekly, except during recess) and due at the beginning
of the next class. It will consist of exercises and practice questions
intended to test and affirm your knowledge of the course content.
While collaboration on the assignments is accepted on the conceptual
level, you must write your solutions strictly individually. Late
homework will be penalized at a rate of 10% per day.
 Exams will include two midterms (1h15 each) during the
semester (see Schedule for tentative dates)
and one final. Makeup exams will not be given except in cases of
documented emergency.
 Important dates:
 Last day to add/drop courses → Jan 24
 Midterm Exam 1 → Feb 26 (tentative)
 Last day to withdraw → Apr 2
 Midterm Exam 2 → Apr 9 (tentative)
 Final Exam → May 7, 5:457:45pm
Both grading policy and scale are subject to change.
Failure (< 70) in either the practice component
(homework) or the lecture component (exams)
will result in an F for the whole course.
• Grading Policy
Homework  40% 
Midterm Exam 1  20% 
Midterm Exam 2  15% 
Final Exam  25% 

⇨

• Grading Scale
100 → 90  A, A 
89 → 80  B+, B, B 
79 → 70  C 
69 → 0  F 

This is a tentative schedule, subject to
readjustment depending on the time we actually spend
in class covering the topics. Please check again regularly.
Course notes (if any), assignments and exam solutions are posted on the
CUA Blackboard,
under "...ENGR 520/Course Documents" and "...ENGR 520/Assignments".
Permanent reading assignment: it is
assumed that you are familiar with the contents of the course notes
of all past meetings.
Week 
Dates 
Topics 
1 
Wed, Jan 15 
1. FirstOrder ODEs:
• introduction: modeling, classification
• separable equations → direct integration
• ax+by+c substitution → reduction to separable
• exact equations → total differential

2 
Wed, Jan 22 
• inexact equations → integrating factor in x, in y, in x^{m}y^{n}
• balanced equations → function of y/x
• linear equations → homogeneous, nonhomogeneous
• Bernoulli equation → reduction to linear

3 
Wed, Jan 29 
2. SecondOrder Linear ODEs:
• homogeneous → "reduction of order" from one solution
• homogeneous with cst coeffs → exponential solutions
• nonhomogeneous with cst coeffs → "undetermined coefficients"

4 
Wed, Feb 5 
• nonhomogeneous → "variation of the constant", Wronskian
• general (nonlinear), missing y, missing x → reduction to firstorder
3. Linear PDEs:
• wave equation (Laplace equation, heat equation)
• "separation of variables" in Cartesian coordinates → planar waves

5 
Wed, Feb 12 
• Laplacian in cylindrical coordinates
• wave equation in cylindrical coordinates → Bessel equations
4. PowerSeries Solutions:
• infinite series → ratio test, root test
• power series → radius of convergence
• solutions of ODEs → index shifting, recurrence relationships

6a 
Mon, Feb 17 12:303pm 3:306pm 
• wave equation in spherical coordinates → Legendre equations
• power series solutions → Legendre polynomials
• singular points, Frobenius method → Bessel solutions
5. Real Definite Integrals:
• Gamma function, Beta function
• Gaussian integral, attenuating factor
• differentiating under the integral

6b 
Wed, Feb 19 
6. Complex Analysis:
• complex numbers, polar form, complex roots
• complex derivative, analytic functions

Midterm Review 1

7 
Wed, Feb 26 
Midterm Exam 1 (1h15)

• CauchyRiemann equations
• complex trigonometric functions, complex logarithm

8 
Wed, Mar 5 
Class cancelled

9 
Wed, Mar 12 
Spring recess  no class

10 
Wed, Mar 19 
7. Complex Integration:
• path integral, contour integral
• Cauchy's integral theorem
• Cauchy's integral formula

11 
Wed, Mar 26 
8. Complex Series:
• Taylor series, Laurent series
• residue integration theorem
• application to real integrals

12 
Wed, Apr 2 
9. Fourier Series:
• period 2π, period 2L
• even and odd functions

Midterm Review 2

13 
Wed, Apr 9 
Midterm Exam 2 (1h15)

• complex expression
• solving forced oscillations

14 
Wed, Apr 16 
10. Fourier Transform:
• Fourier integral
• cosine and sine transforms
• Fourier transform

15 
Wed, Apr 23 
11. Laplace Transform:
• basic functions, step, impulse
• shift, derivative, integral
• solving ODEs algebraically

16 
Wed, Apr 30 
Final Review

17 
Wed, May 7 5:457:45pm 
Final Exam (2h)

