MATH 3113 - Introduction to Ordinary Differential Equations, Section 001 - Spring 2006
MWF 12:30 - 1:20 p.m., 115 PHSC

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu.

Office Hours: MWF 2-3 p.m., or by appointment, in 802 PHSC.

Prerequisite: MATH 2443 (Calculus and Analytic Geometry IV) or concurrent enrollment.

Text: C. H. Edwards, D. E. Penney. Differential Equations and Boundary Value Problems,
3rd ed, Pearson/Prentice Hall, 2004, ISBN 0-13-065245-8.
The course will cover parts of Chapters 1, 3-7.

Homework (Solutions are deposited after the due date in the Chemistry-Mathematics Library, 207 PHSC)

• Homework 1, due Wed, Feb 1.
• Homework 2, due Wed, Feb 8.
• Homework 3, due Wed, Feb 15.
• Homework 4, due Wed, Mar 1.
• Homework 5, due Wed, Mar 8.
• Homework 6, due Wed, Mar 22.
• Homework 7, due Wed, Mar 29.
• Homework 8, due Wed, Apr 12.
• Homework 9, due Wed, Apr 19.
• Homework 10, due Wed, Apr 26.
• Homework 11, due Mon, May 1. NOTE THE UNUSUAL DUE DATE!

Content of the lectures:

• Lecture 1 (Wed, Jan 18): ODEs and mathematical models: Ordinary differential equations (ODEs) of nth order, initial conditions (ICs), initial value problem (IVP), general solution of an ODE, solution of an IVP, general idea of mathematical modeling of natural phenomena, examples (Sec. 1.1).
• Lecture 2 (Fri, Jan 20): Integrals as general and particular solutions: general and particular solutions, examples, position, velocity and acceleration, Newton's Second Law, free vertical motion. Slope fields and solutions curves: geometric meaning of a first-order differential equation, slope field (direction field), examples. (Sec. 1.2, beginning of 1.3).
• Lecture 3 (Mon, Jan 23): Slope fields and solutions curves (cont.): more on slope fields, plotting slope fields with Mathematica. Existence and uniqueness of solutions of IVPs - theorem and examples. (Sec. 1.3).
• Lecture 4 (Wed, Jan 25): Existence and uniqueness of solutions of IVPs (cont.): more examples. Separable equations and applications: Separable equations, solving a separable equation, examples, implicit solutions (Sec. 1.3, 1.4).
• Lecture 5 (Fri, Jan 27): Separable equations and applications (cont.): singular solutions; applications of separable equations - natural growth and decay (population dynamics, compound interest, radioactive decay, drug elimination), half-life of an radioactive isotope, carbon dating, Newton's law of cooling; emptying of a water tank (read it yourself). Linear first-order equations: integrating factor (Sec. 1.4, 1.5).
• Lecture 6 (Mon, Jan 30): Linear first-order equations: remarks about the solutions of linear 1st order ODEs and the IVPs for them, examples, setting up a problem of mixing solutions in a tank (Sec. 1.5).
• Lecture 7 (Wed, Feb 1): Linear first-order equations (cont.): solving the problem of mixing solutions in a tank in some particular cases, reality check (checking that the solution obtained matches our expectations about its behavior). Substitution methods and exact equations: substitution method, examples (Sec. 1.5, 1.6).
• Lecture 8 (Fri, Feb 3): Homogeneous equations: basic substitution converting them to separable equations, examples (Sec. 1.6).
• Lecture 9 (Mon, Feb 6): Bernoulli equations: basic substitution converting them to linear equations, examples. Exact equations: introduction (Sec. 1.6).
• Lecture 10 (Wed, Feb 8): Exact equations (cont.): method of solution, examples. Reducible second-order equations: equations with the dependent variable y missing, equations with the independent variable x missing (Sec. 1.6).
• Lecture 11 (Fri, Feb 10): Population models: logistic (Verhulst) equation - carrying capacity, solving the equation, asymptotic behavior of its solutions (Sec. 2.1).
• Lecture 12 (Mon, Feb 13): Population models (cont.): extinction-explosion equation - behavior of its solutions. Equilibrium solutions and stability Newton's law of cooling - behavior of the solutions; autonomous first order ODEs, equilibrium solutions, connection between equilibrium solutions and critical points of the right-hand side of the ODE, example (logistic equation); stability of critical points - stable and unstable critical points (Sec. 2.2).
• Lecture 13 (Wed, Feb 15): Review for Hour exam 1.
• Lecture 14 (Fri, Feb 17): Hour exam 1.
• Lecture 15 (Mon, Feb 20): Equilibrium solutions and stability (cont.): more on stable and unstable equilibrium solutions, examples of finding the equilibrium solutions and determining their stability (Sec. 2.2).
• Lecture 16 (Wed, Feb 22): Equilibrium solutions and stability (cont.): logistic population with harvesting - analysis of the equilibrium solutions, bifurcation diagram (Sec. 2.2). Second-order linear equations: definition of a linear equation, homogeneous and nonhomogeneous linear equations, principle of superposition for homogeneous linear equations (Sec. 3.1).
• Lecture 17 (Fri, Feb 24): Second-order linear equations (cont.): existence and uniqueness of solutions of 2nd order linear equations, examples; linearly dependent and linearly independent solutions of a 2nd order linear equation, examples; Wronskian of two functions, relationship between the value of the Wronskian and the linear dependence of two solutions of a 2nd order linear equation, examples (Sec. 3.1).
• Lecture 18 (Mon, Feb 27): Second-order linear equations (cont.): general solutions of homogeneous 2nd order linear equation, examples; homogeneous 2nd order linear equations with constant coefficients - characteristic equation, general solution of the equation in the case of two distinct real roots of the characteristic equation and in the case of one (real) root of the characteristic equation (Sec. 3.1).
• Lecture 19 (Wed, Mar 1): General solutions of linear equations: nth order linear differential equation and the associated homogeneous equation, writing the equations as Ly=f(x), resp. Ly=0 (see page 166 for the definition of L); principle of superposition for homogeneous equations, existence and uniqueness for linear equations, linear dependence of functions, linearly independent solutions, Wronskian of functions, relationship between linear dependence of solutions y₁, ..., y_n, and the vanishing of their Wronskian W(y₁,...,y_n), general solutions of linear homogeneous equations (Sec. 3.2).
• Lecture 20 (Fri, Mar 3): General solutions of linear equations (cont.): general solutions of linear nonhomogeneous equations. Homogeneous linear equations with constant coefficients: characteristic equation; case 1 - distinct real roots of the characteristic equation (Sec. 3.3).
• Lecture 21 (Mon, Mar 6): Homogeneous linear equations with constant coefficients (cont.): polynomial differential operators L=p_n(x)Dⁿ+p_n-1(x)D^n-1+...+p₁(x)D+p₀(x), commutativity of the pair of operators (D-a) and (D-b) for a and b constants, examples of non-commuting operators; case 2 - repeated roots of the characteristic equation: solving the equation (D-r₁)^ky=0 for k>1, examples; imaginary numbers, exponent of an imaginary number, Euler's formula, complex numbers, exponents of complex numbers, geometric representation of complex numbers, polar coordinates (Sec. 3.3).
• Lecture 22 (Wed, Mar 8): Homogeneous linear equations with constant coefficients (cont.): complex conjugate, facts about taking complex conjugates of sums and products of complex numbers, proof of the fact that the complex zeros of a polynomial with real coefficients come in conjugate pairs; contribution to the general solution due to a conjugate pair of complex roots, each with multiplicity k, examples (Sec. 3.3).
• Lecture 23 (Fri, Mar 10): Nonhomogeneous second order linear equations with constant coefficients:
  • (general solution of nonhomogeneous equation) = (general solution of homogeneous equation) + (particular solution of nonhomogeneous equation);
  • if f(x)=f₁(x)+f₂(x), and if y_p,1(x) and y_p,2(x) are particular solutions of Ly=f₁(x) and Ly=f₂(x), resp., then y_p(x)=y_p,1(x)+y_p,2(x) is a particular solution of Ly=f(x);
  • finding a particular solution of Ly=f(x) in the case f(x)=e^cxP_n(x): if c is a root of the characteristic equation of multiplicity s, then look for a particular solution of the nonhomogeneous equation of the form y_p(x)=x^se^cxQ_n(x);
  • finding a particular solution of Ly=f(x) in the case f(x)=e^cxP_n(x)cos(dx) or f(x)=e^cxP_n(x)sin(dx): if c+id is a root of the characteristic equation of multiplicity s, then look for a particular solution of the nonhomogeneous equation of the form y_p(x)=x^se^cx[Q_n(x)cos(dx)+ R_n(x)sin(dx)];
  • for more details see my notes (in pdf format) on finding general solutions of homogeneous and nonhomogeneous linear equations with constant coefficients
  (Sec. 3.5).

• Lecture 24 (Mon, Mar 20): Nonhomogeneous second order linear equations with constant coefficients (cont.): variation of parameters, example (Sec. 3.5).
• Lecture 25 (Wed, Mar 22): Mechanical vibrations: derivation of the differential equation of free (or forced) damped oscillations, simple pendulum (derivation from the law of conservation of energy), free undamped motion - amplitude, angular (circular) frequency, period, linear frequency; free damped motion - overdamped, critically damped, and underdamped cases (Sec. 3.4).
• Lecture 26 (Fri, Mar 24): Forced oscillations and resonance: differential equation of forced damped mechanical oscillations, undamped forced oscillations (non-resonant and resonant cases), damped forced oscillations (transient and "steady" periodic parts of the general solution), differential equation of electrical circuits (general solution, adjusting the resonant frequency Ω_res=(LC)^-1/2 by changing the capacitance in the circuit) (Sec. 3.6, 3.7).
• Lecture 27 (Mon, Mar 27): Endpoint problems and eigenvalues: examples of existence and non-existence of non-trivial solutions of BVPs for linear homogeneous second-order equations, general setup (for 2nd order linear ODEs) - looking for eigenvalues and eigenfunctions of the BVP y''+p(x)y'+λq(x)y=0, y(0)=0, y(L)=0; computing the eigenvalues and eigenfunctions of the problem y''+λy=0, y(0)=0, y(L)=0 (Sec. 3.8).
• Lecture 28 (Wed, Mar 29): Review for Hour exam 2.
• Lecture 29 (Fri, Mar 31): Hour exam 2.
• Lecture 30 (Mon, Apr 3): Laplace transforms and inverse transforms: definition of Laplace transform (LT), examples (LTs of f(t)=1, f(t)=e^at), Gamma function, LT of f(t)=t^a; properties of LTs - linearity (Theorem 1), existence (Theorem 2), behavior for s→∞ (Theorem 3) (Sec. 7.1).
• Lecture 31 (Wed, Apr 5): Laplace transforms and inverse transforms (cont.): examples of operators, LP of a unit step function u_a(x)=u(x-a). Transformation of initial value problems: transforms of derivatives (Theorem 1 and Corollary), solution of IVPs, examples (Sec. 7.1, 7.2).
• Lecture 32 (Fri, Apr 7): Transformation of initial value problems (cont.): additional transform techniques, examples; transforms of integrals (Theorem 2), examples. Translations and partial fractions: partial fractions - Rules 1 and 2, example (Sec. 7.2, 7.3).
• Lecture 33 (Mon, Apr 10): Translations and partial fractions (cont.): Laplace transforms of functions with finite jump discontinuities (p. 455); translation on the s-axis, examples of application (Sec. 7.3).
• Lecture 34 (Wed, Apr 12): Translations and partial fractions (cont.): dealing with high powers of terms in the denominators in the method of partial fractions, general idea of integral transforms (Sec. 7.3).
• Lecture 35 (Fri, Apr 14): Derivatives, integrals, and product of transformations: convolution of two functions, example, Laplace transform of a convolution (Theorem 1), differentiation of a Laplace transform (Theorem 2), integration of transforms (Theorem 3) (Sec. 7.4).
• Lecture 36 (Mon, Apr 17): Impulses and delta functions: delta function as a limit of a "rectangular" functions f_a,ε(t) as ε→0, Laplace transform of a delta function, delta function inputs, example, systems analysis and Duhamel's principle (Sec. 7.6).
• Lecture 37 (Wed, Apr 19): First-order systems and applications: first-order systems, a solution of a first-order system, writing a single nth-order ODE as a first-order system of n equations, examples, simple 2-dimensional systems, examples, homogeneous and nonhomogeneous linear systems (Sec. 4.1).
• Lecture 38 (Fri, Apr 21): Matrices and linear systems: definition of a matrix A=(a_i,j), zero matrix 0, unit matrix I, addition of matrices A+B=(a_i,j+b_i,j), multiplication by a number cA=(ca_i,j), transpose (A^T)_i,j=(a_j,i), matrix multiplication, properties of multiplication (NOTE: AB≠BA in general!), inverse of a matrix, matrix-valued functions A(t)=(a_i,j(t)), derivative of a matrix-valued function A'(t)=(a'_i,j(t)) (Sec. 5.1).
• Lecture 39 (Mon, Apr 24): Matrices and linear systems: first-order linear systems (homogeneous and nonhomogeneous) written in matrix form, principle of superposition for homogeneous linear systems, independence and general solutions, Wronskian of solutions, solutions of nonhomogeneous systems. The eigenvalue method for homogeneous systems: eigenvalues and eigenvectors, condition for existence of nontrivial solution of a linear system (Sec. 5.1, 5.2).
• Lecture 40 (Wed, Apr 26): The eigenvalue method for homogeneous systems (cont.): algorithm for finding eigenvalues and eigenvectors (in the case when all eigenvalues are distinct) (Sec. 5.2).
• Lecture 41 (Fri, Apr 28): The eigenvalue method for homogeneous systems (cont.): solving a homogeneous linear system in the case of distinct real eigenvalues (Sec. 5.2).

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also strongly encouraged. Pop-quizzes will be given in class at random times, and all pop-quiz grades will be taken into account in forming your final grade. You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or phone call from a doctor or a parent).

Homework: Homework assignments will be set regularly throughout the semester and will be posted on this web-site. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. Your lowest homework grade will be dropped. All hand-in assignments must be submitted in class on the due date. No late homework will be accepted!

Shortly after a homework assignment's due date, solutions to the problems from that assignment will be placed on restricted reserve in the Chemistry-Mathematics Library in 207 PHSC.

Grading: Your grade will be determined by your performance on the following coursework:

• three midterm exams, 50 minutes each (each midterm contributes 17% to your overall grade);
• homework (20% of your overall grade);
• pop-quizzes (5% of your overall grade);
• final exam (24% of your overall grade).

Good to know:

• The greek_alphabet.

Some Important Dates :

1. Final Day to Register or Add a Class: Friday, January 20.
2. Last day to drop a class with a refund: Monday, January 30.
3. Last day to drop a class without recorded grade: Monday, January 30.
4. Last day to withdraw with an automatic W: Friday, February 24.
5. Last day to withdraw with a W/F without permission of the Dean: Friday, May 5.

Policy on W/I Grades : Through Friday, February 24, you can withdraw from the course with an automatic W. In addition, it is my policy to give any student a W grade, regardless of his/her performance in the course, through the extended drop period that ends on Friday, May 5. However, after May 5, you can only drop via petition to the Dean of your college. Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition. Thus it is in your own best interest to drop the course on or before May 5 if you think there is a reasonable chance that you will not want to see the course through to the end.

The grade of I (Incomplete) is not intended to serve as a benign substitute for the grade of F. I only give the I grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it! For more details on the University's policies concerning academic misconduct click here.

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD (only) 405-325-4173.