MATH 3413 - Physical Mathematics I, Section 001 - Spring 2006
TR 1:30 - 2:45 p.m., 114 PHSC

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

Office Hours: Mon 2-3 p.m., Tue 11 a.m.-noon, Wed 1:30-2:30 p.m., or by appointment.

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.

Date for the Final Exam: Thu, Dec 14, 1:30-3:30 p.m.

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

• Homework 1, due Thu, Aug 31.
• Homework 2, due Thu, Sep 7.
• Homework 3, due Thu, Sep 14.
• Homework 4, due Thu, Sep 21.
• Homework 5, due Thu, Oct 5.
• Homework 6, due Thu, Oct 12.
• Homework 7, due Thu, Oct 19.
• Homework 8, due Tue, Oct 24, in class! Note the unusual due date!
• Homework 9, due Thu, Nov 2.
• Homework 10, due Thu, Nov 9.
• Homework 11, due Tue, Nov 21.
• Homework 12, due Tue, Nov 28.
  A solution of Additional problem 1 (the same I gave in class).

Course content:

• Separable equations, linear equations, applications.
• Homogeneous, Bernoulli, exact equations.
• Existence and uniqueness for first order ODEs. Numerical methods.
• Second order nonhomogeneous equations. Variations of parameters. Mass-spring system, resonance.
• First order systems.
• Laplace transform and applications to ODEs. Delta function.
• Power series method. Legendre and Bessel functions.
• Fourier series.
• Heat conduction problem with Dirichlet and Neumann boundary conditions.
• String vibration problems.
• Laplace equation on a rectangle.
• Problems in circular and cylindrical regions.

Content of the lectures:

• Lecture 1 (Tue, Aug 22): Differential equations and mathematical models: examples, generalities about the process of mathematical modeling (Sec. 1.1).
  Integrals as general and particular solutions: general solution of an ODE, initial conditions (ICs), initial value problems (IVPs), particular solution of an IVP, examples (Sec. 1.2).
• Lecture 2 (Thu, Aug 24): Separable equations and applications: separable equations - method of solution, examples; implicit solutions and singular solutions - examples; applications - natural growth and decay, radioactive decay, half-life of an isotope, ¹⁴C dating, drug elimination, Newton's law of cooling (Sec. 1.4).
  Linear first-order equations: method of solution, integrating factor ρ(x), examples (Sec. 1.5).
• Lecture 3 (Tue, Aug 29): Linear first-order equations (cont.): more examples, Si(x) function (integral sine) in a solution of an IVP, mixture problems (Sec. 1.5).
  Substitution methods and exact equations: substitution methods - idea, example; homogeneous equations - definition, method of solution, example; Bernoulli equation - definition, idea of solution (Sec. 1.6).
• Lecture 4 (Thu, Aug 31): Substitution methods and exact equations (cont.): Bernoulli equation - example; implicitly defined solutions F(x,y(x))=0, examples, deriving the ODE satisfied by an implicitly defined solution; exact equations, method of solution, examples, fragility of the "exactness" property (Sec. 1.6).
• Lecture 5 (Tue, Sep 5): Substitution methods and exact equations (cont.): more examples of exact equations; equations with dependent variable (y) missing, generalization of the idea, examples; equations with the independent variable (x) missing, examples (Sec. 1.6).
  Slope fields and solution curves: slope fields and solution curves of ODEs, general remarks about the behavior of the solution curves, idea of singular solutions (Sec. 1.3).
• Lecture 6 (Thu, Sep 7): Second-order linear equations: a general definition of an n^th order linear equation, homogeneous and non-homogeneous equations, Principle of Superposition for homogeneous linear equations, existence and uniqueness (briefly), linear independence of two functions, Wronskian of two functions, examples (Sec. 3.1).
• Lecture 7 (Tue, Sep 12): 2nd-order linear equations (cont.): mathematical operations as unary, binary, ternary, ..., "machines", comments about linear and non-linear "machines", examples; finding the general solution of a homogeneous 2nd-order linear equations with constant coefficients: characteristic equation, general solutions in the different cases:
  Case 1 (two distinct real roots r₁≠r₂ of the char. eqn.): y(x)=C₁e^r₁x+C₂e^r₂x;
  Case 2 (one double root r₁ of the char. eqn.): y(x)=(C₁+C₂x)e^r₁x;
  Case 3 (two non-real, necessarily complex conjugate, roots r₁=α+iβ and r₂=α-iβ of the char. eqn.): y(x)=C₁e^αxcos(βx)+C₂e^αxsin(βx);
  idea of complex numbers, Euler's formula e^iθ=cosθ+isinθ, sin and cos in terms of complex exponents, another interesting formula: e^iπ+1=0 (Sec. 3.1, 3.2 - read only the parts about 2nd-order equations).
• Lecture 8 (Thu, Sep 14): 2nd-order linear equations (cont.): derivation of Euler's formula, polar representation of a complex number, complex conjugate and modulus of a complex number (Sec. 3.2, 3.3 - read only the parts about 2nd-order equations).
  Digression: convergent and absolutely convergent series, troubles with reshuffling non-absolutely convergent series ("proof" that ln(2)=0), "Riemann's derangement theorem" about the possibitlity of reshuffling such a series to make its sum equal to any given number.
  Nonhomogeneous 2nd order linear equations with constant coefficients: (general solution of nonhomogeneous equation) = (general solution of homogeneous equation) + (particular solution of nonhomogeneous equation); finding a particular solution of ay'' + by' + cy =f(x) in the case f(x)=e^cxP_m(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_m(x), and find the coefficients of the m^th-degree polynomial Q_m(x) by plugging it in the nonhomogeneous equation; use this method to solve Examples 1, 2, 4 on pages 196-197 (beginning of Sec. 3.5).
• Lecture 9 (Tue, Sep 19): Nonhomogeneous 2nd order linear equations with constant coefficients (cont.): finding a particular solution of ay'' + by' + cy =f(x) in the case f(x)=e^cx[P_m1(x)cos(dx)+R_m2(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_m(x)cos(dx)+T_m(x)sin(dx)], where Q_m(x) and T_m(x) are polynomials of degree m=max(m₁,m₂), and find the coefficients Q_m(x) and T_m(x) by plugging it in the nonhomogeneous equation; use this method to solve Examples 3 and 9 on pages 196-203 (in Sec. 3.5).
  Mechanical oscillations: derivation of the equation of (forced damped) motion, equilibrium, Hooke's law, spring constant, undamped/damped, free/forced motion, linearization of the differential equation of the simple pendulum; free undamped motion, amplitude, angular (circular) frequency, phase, initial phase, linear frequency, period (Sec. 3.4).
• Lecture 10 (Thu, Sep 21): Mechanical oscillations (cont.): free damped motion; forced undamped motion, resonance (Sec. 3.4).
• Lecture 11 (Tue, Sep 26): Hour exam 1.
• Lecture 12 (Thu, Sep 28): Mechanical oscillations (cont.): forced damped motion, resonance in this case, principles of RLC circuits, selectivity of an RLC circuit at resonance (Sec. 3.4).
  First-order systems: definition, conversion of a single nth order ODE into a first-order system, linear systems (homogeneous and non-homogeneous) (Sec. 4.1).
• Lecture 13 (Tue, Oct 3): First-order systems (cont.): more on applications of first-order systems to problems of mechanics (coming from Newton's second law applied to systems with one degree of freedom), phase space (Sec. 4.1).
  Laplace transforms (LTs) and inverse transforms: definition of LT, LT as a unary "machine", LT of f(t)=1, f(t)=e^at, Gamma function and its properties, LT of f(t)=t^a, linearity of LT, applications of the linearity property, LT of the unit step function u_a(t)=u(t-a), (very briefly on the) existence and uniqueness of LT and inverse LT (Sec. 7.1).
  LT of initial value problems: LT of derivatives, solving an IVP by using LT (pages 446-449 of Sec. 7.2).
• Lecture 14 (Thu, Oct 5): LT of initial value problems (cont.): LT of integrals, an application (Sec. 7.2).
  Translation and partial fractions: rules for partial fractions, translation on the s-axis, examples (Sec. 7.3).
  Derivatives, integrals, and products of transforms: convolution of two functions (Sec. 7.4).
• Lecture 15 (Tue, Oct 10): Derivatives, integrals, and products of transforms: commutativity property of convolution (f*g=g*f), Theorem 1 (the convolution property), proof of the theorem, examples of application, Theorem 2 (differentiation of transforms), proof of the theorem, examples of application, Laplace transform of the Bessel function J₀(t) (Sec. 7.4).
• Lecture 16 (Thu, Oct 12): Derivatives, integrals, and products of transforms(cont.): integration of transforms (Theorem 3) (Sec. 7.4, page 471).
  Periodic and piecewise continuous functions: translation on the t-axis (Theorem 1 on page 475) - LT of u(t-a)f(t-a) (do the proof yourself), examples of application (Sec. 7.5 - pages 475-476).
  Staircase function and its LT: Problem 7.1/39, tricks using the formula for the sum of a geometric series (page 444).
  Impulses and δ functions: motivation of the concept of δ funcion, definition of δ function, integrals involving δ functions, δ functions and step functions: δ_a(t)= u'_a(t) (Sec. 7.6 - pages 486-489, 491).
• Lecture 17 (Tue, Oct 17): Impulses and δ functions (cont.): definition of derivatives of δ_a and intuition behind them; LT of delta function: L{δ_a(t)}=e^-as, L{δ(t)}=1; transfer function W(s) and weight function w(t) of a system, Duhamel's principle: response x(t)=(w*f)(t) for f(t)-driving of the system; measuring w(t) "experimentally": if the driving is f(t)=δ(t), then the response is x(t)=(w*δ)(t)=w(t) (Sec. 7.6 - pages 489-494).
  Introduction and review of power series: power series, convergent series and functions defined by them, examples (Sec. 8.1 - pages 497-499).
• Lecture 18 (Thu, Oct 19): Introduction and review of power series (cont.): binomial coefficients, binomial formula for expanding (1+x)^α for arbitrary α, idea of the power series method, termwise differentiation of power series (Theorem 1), identity principle (Theorem 2), example, radius of convergence (Sec. 8.1 - pages 499-504).
• Lecture 19 (Tue, Oct 24): Remarks on delta function and step function: more on the meaning of δ_a(t) and u_a(t) and the relationship between them.
  Introduction and review of power series (cont.): examples of power series and their radii of convergence, Legendre polynomials for general α>-1, Legendre polynomials for nonnegative integer &alpha (Sec. 8.1 - pages 506-509; Sec. 8.2 - pages 517-519).
• Lecture 20 (Thu, Oct 26): Hour exam 2.
• Lecture 21 (Tue, Oct 31): Periodic functions and trigonometric series: periodic functions, period; idea of Fourier series - expansion of a (periodis) function in terms of sines and cosines; decomposition of vectors with respect of an orthogonal basis: v=Σ_j v_j e_j, where e_j.e_k=0 for j not equal to k; finding the components: v_j=(v.e_j)/(e_j.e_j), defining a "dot product" of periodic functions of period p as an integral, checking that the functions 1, sint, cost, sin2t, cos2t, sin3t, cos3t,... are orthogonal to one another, writing a periodic function of period p as a linear combination of the functions 1, sint, cost, sin2t, cos2t, sin3t, cos3t,... (Sec. 9.1).
• Lecture 22 (Thu, Nov 2): Periodic functions and trigonometric series (cont.): formulae for the Fourier coefficient of a periodic function of period p=2π, an example (Sec. 9.1).
  General Fourier series and convergence: formulae for the Fourier coefficient of a periodic function of period p=2L;, convergence of a Fourier series (Theorem 1), examples, identities obtained by using Theorem 1 (Sec. 9.2).
• Lecture 23 (Tue, Nov 7): Fourier sine and cosine series: extending a function defined on (0,L) to a periodic function of period 2L as an odd function (Fourier sine series), extending a function defined on (0,L) to a periodic function of period 2L as an odd function (Fourier sine series) (Sec. 9.3).
  Heat conduction and separation of variables: physical meaning of the heat equation and the boundary and initial conditions for it; superposition of solutions u_n(x,t) each of which satisfies the PDE and the BCs, adjusting the coefficients in the superposition in order to satisfy the IC (Sec. 9.5).
• Lecture 24 (Thu, Nov 9): Heat conduction and separation of variables (cont.): finding the solutions u_n(x,t) by separation of variables in the case of Dirichlet BCs u(0,t)=0, u(L,t)=0 (Sec. 9.5).
• Lecture 25 (Tue, Nov 14): Heat conduction and separation of variables (cont.): separation of variables in the case of Neumann BCs u_x(0,t)=0, u_x(L,t)=0, examples (Sec. 9.5).
  Vibrating strings and the one-dimensional wave equation: physical meaning of the wave equation and the boundary and initial conditions for it, separation of variables in the wave equation in the case of Dirichlet BCs u(0,t)=0, u(L,t)=0, imposing the ICs u(x,0)=f(x), u_t(x,0)=g(x) (Sec. 9.6).
• Lecture 26 (Thu, Nov 16): Vibrating strings and the one-dimensional wave equation: separation of variables in the wave equation in the case of Neumann BCs u_x(0,t)=0, u_x(L,t)=0, imposing the ICs u(x,0)=f(x), u_t(x,0)=g(x); representation of a solution of the wave equation in the form u(x,t)=F(x-ct)+G(x+ct), physical meaning; incorporating the effect of air resistance and of the weight of the string in the wave equation (Sec. 9.6).
• Lecture 27 (Tue, Nov 21): Stationary temperature distribution and Laplace equation: physical problems leading to Laplace's equation (digression: sources of heat and Poisson's equation), boundary value problems for 2-dimensional Laplace's equation in a rectangular domain (x,y)∈[0,a]×[0,b], separation of variables in Laplace's equation in the case of Dirichlet BSs u(x,0)=0, u(x,b)=0, u(0,y)=0, u(a,y)=f(y), solution of the Dirichlet boundary value problem Δu=0, u(x,0)=0, u(x,b)=f(x), u(0,y)=0, u(a,y)=0 by analogy (Sec. 9.7).
• Lecture 28 (Tue, Nov 28): Stationary temperature distribution and Laplace equation (cont.): physical meaning of Neumann BCs, solution of the Neumann-Dirichlet boundary value problem Δu=0, u_x(0,y)=0, u_x(a,y)=0, u(x,0)=0, u(x,b)=f(x); solution of Laplace's equation in a circular domain - periodic BCs for Θ(θ), consequences for the function R(r) from the requirement of boundedness of the solution u(r,θ) (Sec. 9.7).
• Lecture 29 (Thu, Nov 30): Classes canceled.
• Lecture 30 (Tue, Dec 5): Hour exam 3.
• Lecture 31 (Thu, Dec 7): Separation of variables in cylindrical geometry: solving the heat equation in an infinite cylinder if the temperature distribution depends only on the radial coordinate r and the time t, Bessel equation, Bessel and Neumann functions, discretizing the separation constant coming from zeros of the equation J₀(ξ)=0, orthogonality relations for Bessel functions, determining the constants in the series expansion of u(r,t) from the initial condition by using an orthogonality relation (Sec. 10.4).

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 and 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.

Academic calendar for Fall 2006.

Policy on W/I Grades : Through September 29, 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 December 8. However, after December 8, 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 December 8 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 see http://www.ou.edu/provost/integrity/.

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.