MATH 3113.001 - Introduction to Ordinary Differential Equations - Fall 2013
MWF 1:30 - 2:20 p.m., 122 PHSC

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

Office Hours: Mon 2:30-3:30 p.m., Wed 10:15-11:15 a.m., or by appointment, in 802 PHSC.

Please take a couple of minutes to fill out your evaluation of the course!
Here is a link to the evaluation web-site: http://eval.ou.edu; it closes on December 8th (Sunday).

Course catalog description: Prerequisite: MATH 2423 or MATH 2924. Duplicates two hours of 3413. First order ODEs, linear differential equations with constant coefficients, two-by-two linear systems, Laplace transformations, phase planes and stability. (F, Sp, Su)

Text: C. H. Edwards, D. E. Penney. Differential Equations and Boundary Value Problems,
4th ed, Pearson/Prentice Hall, 2008, ISBN 978-0-13-235658-9.

Check out the OU Math Blog! It is REALLY interesting!

Check out the Problem of the Month!

Homework:

• Homework 1 (problems given on August 19, 21, 23), due August 30 (Friday).
• Homework 2 (problems given on August 26, 28, 30), due September 6 (Friday).
• Homework 3 (problems given on September 4, 6, 9), due September 16 (Monday).
• Homework 4 (problems given on September 11, 13, 16), due September 25 (Wednesday). Please note the new due date!
• Homework 5 (problems given on September 20, 23, 25), due October 2 (Wednesday).
• Homework 6 (problems given on September 27, 30, October 2), due October 9 (Wednesday).
• Homework 7 (problems given on October 4, 7, 9), due October 16 (Wednesday).
• Homework 8 (problems given on October 14, 16, 21, 23), due October 30 (Wednesday).
• Homework 9 (problems given on October 25, 28, 30, November 1), due on November 8 (Friday).
• Homework 10 (problems given on November 4, 6, 8), due on November 15 (Friday).
• Homework 11 (problems given on November 11, 13, 15, 20), due on November 25 (Monday). Please note the new due date!
• Homework 12 (problems given on November 22, 25, December 2), due on December 6 (Friday).

Content of the lectures:

• Lecture 1 (Mon, Aug 19): 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, mathematical modeling of natural phenomena, examples [Sec. 1.1].
  Homework: Problems 1.1/6, 15, 23, 31, 35, 38.
  FFT: Problems 1.1/29, 42.
  Remark: The FFT ("Food For Thought") problems are to be solved like a regular homework problems, but do not have to be turned in.
• Lecture 2 (Wed, Aug 21): Integrals as general and particular solutions: general and particular solutions, examples, position, velocity and acceleration, Newton's Second Law, free vertical motion [Sec. 1.2]
  Slope fields and solutions curves: geometric meaning of a first-order differential equation, slope field (direction field), plotting slope fields with Mathematica; existence and uniqueness of solutions of IVPs: an example of and IVP with no solution, an example of and IVP with infinitely many solutions (growth of the volume of a water droplet in an oversaturated vapor - see Problem 1.3/29) [Sec. 1.3]
  Homework: Problems 1.2/5, 6, 24, 35; 1.3/29.
  FFT: Problems 1.2/10, 20; 1.3/27.
• Lecture 3 (Fri, Aug 23): Existence and uniqueness of solutions of IVPs (cont.): Theorem on the existence and uniqueness of solutions of IVPs for ODEs, discussion of the "bad" examples from Lecture 2 in light of the Theorem; an example of an IVP whose solution does not exist for all x>0 [Sec. 1.3]
  Separable equations and applications: separable equations, solving a separable equation, examples, implicit solutions [pages 32-36 of Sec. 1.4]
  Homework: Problems 1.4/5, 13, 21, 25, 28.
  FFT: Problems 1.4/17, 30.
  The complete Homework 1 is due on August 30 (Friday).
• Lecture 4 (Mon, Aug 26): Separable equations and applications (cont.): singular solutions; applications of separable equations - natural growth and decay: population dynamics, logistic equation, radioactive decay, carbon dating, Newton's law of cooling; emptying of a water tank (please read it yourself) [pages 36-42 of Sec. 1.4]
  Linear first-order equations: integrating factor, algorithm for solving linear first-order equations [pages 48-51 of Sec. 1.5]
  Homework: Problems 1.4/35, 49; 1.5/4, 14, 17, 23, 27, 29.
  FFT: Problems 1.4/61, 1.5/31.
• Lecture 5 (Wed, Aug 28): Linear first-order equations (cont.): remarks on the existence and uniqueness of solutions of IVPs for linear first-order equations; 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) [pages 51-55 of Sec. 1.5]
  Substitution methods and exact equations: substitution method, examples; homogeneous equation: substitution converting a homogeneous equation to a separable equations, examples [pages 60-64 of Sec. 1.6]
  Homework: Problems 1.5/31; 1.6/9, 12, 17, 18, 57.
  FFT: Problems 1.6/55.
• Lecture 6 (Fri, Aug 30): Substitution methods and exact equations (cont.): homogeneous equation - more examples; Bernoulli equation - a substitution that reduces it to a linear first order equation; implicitly and explicitly given solutions, deriving an equation from its solution (given implicitly), exact equations [pages 64-69 of Sec. 1.6]
  Homework: Problems 1.6/8, 19, 23, 28 [set v(x)=e^y(x)], 29 [set v(x)=sin(y(x))].
  FFT: Problems /63.
  The complete Homework 2 is due on September 6 (Friday).
• Lecture 7 (Wed, Sep 4): Substitution methods and exact equations (cont.): exact equations: a method for solving them, "fragility" of the exactness property; equations with the dependent variable y missing [pages 70-72 of Sec. 1.6]
  Homework: click on the link to download the assigned homework.
• Lecture 8 (Fri, Sep 6): Substitution methods and exact equations (cont.): equations with the independent variable x missing [page 73 of Sec. 1.6]
  Equilibrium solutions and stability: autonomous first order ODEs, equilibrium solutions, connection between equilibrium solutions and critical points (zeros) of the right-hand side of the ODE, example (logistic equation); stable and unstable equilibria, examples of finding the equilibrium solutions and determining their stability [pages 92-96 of Sec. 2.2]
  Homework: click on the link to download the assigned homework.
• Lecture 9 (Mon, Sep 9): Equilibrium solutions and stability (cont.): logistic equation with harvesting: dP/dt=αP(1−P/K)−H, determining the units of the quantities; change of variables to reduce the number of parameters: dx/dt=αx(1−x)−h (where x=P/K is the "dimensionless" population, and h=H/K), studying the number and types of equilibrium solutions of this equation: two equilibrium solutions when 0<h<α/4 (the smaller one repelling, the large one attracting), one semi-stable equilibrium solution when h=α/4, zero equilibrium solutions when h>α/4 (over-harvesting), representing roughly the time evolution of the system (in the (t,x)-plane), bifurcation diagram in the (h,α)-plane; reducing the number of parameters to one by choosing new "time" variable, τ:=αt, to obtain dx/dτ=x(1−x)−μ [pages 97-98 of Sec. 2.2]
  Homework: click on the link to download the assigned homework.
  The complete Homework 3 is due on September 16 (Monday).
• Lecture 10 (Wed, Sep 11): Equilibrium solutions and stability (cont.): plotting the equilibrium solutions of dx/dt=αx(1−x)−h in the (h,x)-plane for a fixed positive value of α (stable equilibria with a solid line, unstable equilibria with a dashed line) [page 98 of Sec. 2.2]
  Second-order linear equations: definition of a linear equation, homogeneous and nonhomogeneous linear equations, homogeneous equation associated with a nonhomogeneous equation; a physical example leading to a second-order linear equation: oscillator with resistance force and external driving [pages 147-149 of Sec. 3.1]
  Homework: Problems 2.2/29; 3.1/7, 17, 19.
• Lecture 11 (Fri, Sep 13): Second-order linear equations (cont.): Principle of Superposition for homogeneous linear equations; theorem on existence and uniqueness of solutions of 2nd order linear equations; linearly dependent and linearly independent functions; Wronskian of two functions; constructing the general solution of a homogeneous second order linear equation as a linear combination of two solutions of the equation with nonzero Wronskian [pages 150-153 of Sec. 3.1]
  Homework: Problems 3.1/10, 20, 24, 25, 27, 28, 29.
  FFT: Problem 3.1/32.
• Lecture 12 (Mon, Sep 16): Second-order linear equations (cont.): the Wronskian of two solutions of a second-order linear ODE is either identically zero or never becomes zero; 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]
  Homework: Problems 3.1/33, 35, 39, 43, 45, 47, 48.
  The complete Homework 4 is due on September 25 (Wednesday).
• Lecture 13 (Wed, Sep 18): Exam 1 [on the material from Sec. 1.1-1.6, 2.2, covered in Lectures 1-9 and in the first half of Lecture 10]
• Lecture 14 (Fri, Sep 20): 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; principle of superposition for homogeneous linear equations; existence and uniqueness theorem for solutions of linear equations; linear dependence of functions; 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 of order n (a superposition of n linearly independent solutions); general solutions of linear nonhomogeneous equations (a sum of the general solution of the associated homogeneous and one solution of the nonhomogeneous equation [Sec. 3.2]
  Homework: Problems 3.1/51, 52; 3.2/2, 5, 8, 17, 23, 25.
• Lecture 15 (Mon, Sep 23): Homogeneous linear equations with constant coefficients: characteristic equation; case 1 - distinct real roots of the characteristic equation; notation D:=d/dx, polynomial differential operators with constant coefficients L=a_nDⁿ+a_n-1D^n-1+...+a₁D+a₀, commutativity of the pair of operators (D-a) and (D-b) for a and b constants (i.e., (D-a)(D-b)=(D-b)(D-a)) - check this directly; case 2 - repeated roots of the characteristic equation: solving the equation (D-r)^ky=0 for k>1, examples [pages 173-178 of Sec. 3.3]
  Homework: click on the link to download the assigned homework.
• Lecture 16 (Wed, Sep 25): Homogeneous linear equations with constant coefficients (cont.): complex numbers, algebraic operations with complex numbers; exponent of a complex number, Euler's formula e^iθ=cosθ+isinθ, sine and cosine functions expressed in terms of complex exponents (derivation from the Euler's formula); 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 [pages 178-180 of Sec. 3.3]
  Homework: Problems 3.3/8, 16, 23, 40, 41, 42.
  The complete Homework 5 is due on October 2 (Wednesday).
• Lecture 17 (Fri, Sep 27): Homogeneous linear equations with constant coefficients (cont.): geometric representation of complex numbers, polar coordinates, complex conjugate; more examples of finding general solutions of homogeneous linear equations with constant coefficients, writing the homogeneous linear equations with constant coefficients whose general solution is given [pages 181-182 of Sec. 3.3]
  Nonhomogeneous equations and undetermined 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);
  For more details see my notes (in pdf format) on finding general solutions of homogeneous and nonhomogeneous linear equations with constant coefficients.
  Homework: Problems 3.5/9, 22, 24, 29, 32.
  FFT: Problems 3.5/43(a).
• Lecture 18 (Mon, Sep 30): Nonhomogeneous second order linear equations with constant coefficients (cont.):
  • 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]
  Homework: click on the link to download the assigned homework.
  FFT: Problems 3.5/43(b), 44.
• Lecture 19 (Wed, Oct 2): Mechanical vibrations: Hooke's law - derivation by Taylor expansion of the potential energy around a nondegenerate local minimum; derivation of the differential equation of free (or forced) damped oscillations x''+2px'+ω₀²x=0 (x''+2px'+ω₀²x=f(t), respectively); free undamped motion - amplitude, angular frequency ω₀, period T=2π/ω₀, linear frequency f=1/T=ω₀/2π; free damped motion - overdamped, critically damped, and underdamped cases [Sec. 3.4]
  Homework: Problems 3.4/30, 31, 32, 33 (the notations used in these problems are defined before Problem 3.4/24); the binomial series needed in Problem 31 is written as Equation (12) on page 505 of the book.
  The complete Homework 6 is due on October 9 (Wednesday).
• Lecture 20 (Fri, Oct 4): Forced oscillations and resonance: differential equation of forced damped mechanical oscillations, damped forced oscillations - transient (tending to zero as t→∞) 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, pages 225-227 of Sec. 3.7]
  Homework: Problems 3.6/25, 27.
• Lecture 21 (Mon, Oct 7): Laplace transforms and inverse transforms: operators - a rule taking a function and producing another function; definition of Laplace transform (LT), examples: LTs of f(t)=1, f(t)=e^at, definition and basic properties of the Gamma function [Γ(x+1)=xΓ(x+1), Γ(n+1)=n!], LT of f(t)=t^a [pages 441-444 of Sec. 7.1, and the table of LTs on page 446]
  Reading assignment: Read Theorem 1 and the three examples after it [pages 444-445 of Sec. 7.1]
  Homework: Problems 7.1/7, 12, 14, 16, 17 [use that cos²α=[1+cos(2α)]/2 for any α], 23, 25, 28.
• Lecture 22 (Wed, Oct 9): Laplace transforms and inverse transforms: inverse Laplace transform; LT of a piecewise continuous functions, example: Laplace transform of the Heaviside function (the unit step function) H_a(x)=H(x−a) (the book uses the notation u_a(x)=u(x−a)); further properties of LTs: existence (Theorem 2), uniqueness of the inverse LT (Theorem 3) [pages 446-450 of Sec. 7.1]
  Transformation of initial value problems: transforms of derivatives (Theorem 1 and Corollary), solution of IVPs, examples [pages 452-455 of Sec. 7.2]
  Homework: Problems 7.1/38, 39; 7.2/2, 7, 25, 27 [hint to Problem 7: you may use that s/[(s²+1)(s²+9)]=(1/8)s/(s²+1)−(1/8)s/(s²+9)].
  The complete Homework 7 is due on October 16 (Wednesday).
• Lecture 23 (Mon, Oct 14): Examples of Laplace transform: a complete solution of Problem 7.1/41 (or 7.2/34) using the formula for the sum of a geometric series; hyperbolic functions - definition and properties of sinh(z), cosh(z), tanh(z), coth(z) [Problem 7.1/41 on page 451 or Problem or 7.2/34 on page 463]
  Homework: Problem 7.2/35.
• Lecture 24 (Wed, Oct 16): Transformation of initial value problems (cont.): transforms of integrals (Theorem 2, with proof), examples [page 460 of Sec. 7.2]
  Translations and partial fractions: translation on the s-axis, examples of application; partial fractions - Rules 1 and 2; example of applying Rule 1 - when the factors in the denominators are of the form (x−a)^k where a is a real number and k is a positive integer [pages 464-468 of Sec. 7.3]
  Homework: Problems 7.2/17, 20, 22; 7.3/2, 5, 6, 7, 14, 15, 28.
• Lecture 25 (Fri, Oct 18): Exam 2 [on the material from Sec. 3.1-3.6, 7.1, 7.2 covered in the second half of Lecture 10 and in Lectures 11, 12, 14-22]
• Lecture 26 (Mon, Oct 21): Translations and partial fractions (cont.): example of applying Rule 2 - when the factors in the denominators are of the form [(x+a)²+b²]^k where a is a real number, b is a positive real number (the positivity of b is not a restrictive condition!), and k is a positive integer; remarks on taking inverse Laplace transform by using partial fraction decomposition and translation on the s-axis; using Mathematica to perform calculations with Laplace transform and its inverse, partial fractions, etc. (use the commands "LaplaceTransform", "InverseLaplaceTransform", "Apart") [pages 465-471 of Sec. 7.3]
  Derivatives, integrals, and product of transformations: definition of convolution of two functions, Laplace transform of the convolution of f(t) and g(t) is a product of the Laplace transforms F(s) and G(s) (Theorem 1, only the statement of the theorem) [pages 474, 475 of Sec. 7.4]
  Homework: Problems 7.3/19, 20, 27, 30, 32; 7.4/1, 6.
  FFT: Problems 7.3/24.
• Lecture 27 (Wed, Oct 23): Derivatives, integrals, and product of transformations: general idea of integral transform (examples: Laplace transform, Fourier transform); Laplace transform of the convolution of f(t) and g(t) is a product of the Laplace transforms F(s) and G(s) (Theorem 1, with proof), examples of applications; differentiation of Laplace transforms (Theorem 2, with proof), examples of applications; [pages 475-478 of Sec. 7.4]
  Reading assignment: read Theorem 3 and the two examples after it [pages 478-479 of Sec. 7.4]
  Homework: click on the link to download the assigned homework.
  The complete Homework 8 is due on October 30 (Wednesday).
• Lecture 28 (Fri, Oct 25): Impulses and delta functions: delta function as a limit of a "rectangular" functions d_a,ε(t) as ε→0 [pages 493-495 of Sec. 7.6]
  Homework: click on the link to download the assigned homework.
• Lecture 29 (Mon, Oct 28): Impulses and delta functions (cont.): more on delta-functions: derivatives of delta-functions: integral of δ_a⁽ⁿ⁾(x)f(x) over the real line is equal to (−1)ⁿf⁽ⁿ⁾(a); Laplace transform of the ODE corresponding to a system driven by an external force f(t), with all initial conditions equal to zero; transfer function W(s) and weight function w(t) corresponding to the system; the Laplace transform X(s) of the response x(t) of the system is equal to the product of the transfer function W(s) and the Laplace transform F(s) of the external driving f(t); the response x(t) of the system is equal to the convolution of the weight function and the external driving: (w*f)(t) [pages 495-497 and 499-500 of Sec. 7.6]
  Homework: click on the link to download the assigned homework.
• Lecture 30 (Wed, Oct 30): Impulses and delta functions (cont.): measuring the weight function w(t) experimentally by applying an external forcing f(t)=δ(t) (or, more accurately, f(t) is equal to the limit of δ_a(t) as a↓0); Duhamel's principle; "proof" that the derivative of the Heaviside (unit step) function u_a(t) is equal to δ_a(t) [pages 498-501 of Sec. 7.6]
  Homework: click on the link to download the assigned homework.
• Lecture 31 (Fri, Nov 1): Impulses and delta functions (cont.): discussion of the mathematical solution and the physical meaning of the problem mx''+cx'+kx=δ_a(t)Δp, x(0)=α, x'(0)=β - the first derivative of x jumps by Δp/m at time a.
  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 [pages 249-250 of Sec. 4.1]
  Matrices and linear systems: definition of a matrix A=(a_ij), addition of matrices A+B=(a_ij+b_ij), multiplication by a number cA=(ca_ij), definition of matrix multiplication [pages 285-288 of Sec. 5.1]
  Homework: click on the link to download the assigned homework.
  The complete Homework 9 is due on November 8 (Friday).
• Lecture 32 (Mon, Nov 4): Matrices and linear systems (cont.): zero matrix, unit matrix (only for square matrices), Kronecker delta δ_ij, transposed matrix, column vector, row vector, inverse matrices, invertible (non-singular) and non-invertible (singular) matrices, uniqueness of the inverse matrix of a given invertible matrix; determinants of 2×2 and 3×3 matrices, Levi-Civita symbol, definition of a determinant of a square matrix by using the Levi-Civita symbol, properties of determinants: det(AB)=det(A)det(B), the determinant of the unit matrix is one, the determinant of A⁻¹ is [det(A)]⁻¹, swapping two rows or swapping two columns of a square matrix flips the sign of the determinant of the matrix [pages 287-291 of Sec. 5.1]
  Homework: Problems 5.1/4, 6, 7, 13, 14.
• Lecture 33 (Wed, Nov 6): Matrices and linear systems (cont.): matrix-valued functions A(t)=(a_ij(t)), derivative of a matrix-valued function A'(t)=(a'_ij(t)), examples; first order linear systems - homogeneous, x'(t)=P(t)x(t), and non-homogeneous, x'(t)=P(t)x(t)+f(t); Principle of Superposition for solutions of homogenous linear systems; linear independence of vectors, examples [pages 291-294 of Sec. 5.1]
  Homework: Problems 5.1/10, 22, 27, 31, 36 [for Problems 22 and 27, for now only verify that the given vectors are solutions of the system - the rest of these problems (finding the Wronskian and writing the general solution) will be a part of the homework assigned in Lecture 34]
  FFT: Some fun links:
  • a video of the Tacoma Narrows Bridge collapse;
  • a video about the Millennium Bridge in London;
  • the NOVA documentary "The Strange New Science of Chaos" (31 January 1989);
  • the Lorenz system of ODEs modeling atmospheric convection;
  • the recent history of chaos theory is discussed in non-technical terms in J. Gleick's book Chaos: Making a new Science (Penguin, 1987);
  • if you would like to learn more about some fascinating phenomena occurring in nonlinear ODEs, check out the course MATH 4193, "Introductory Mathematical Modeling", which will be taught in Spring 2014 by Dr. Alexander Grigo.
• Lecture 34 (Fri, Nov 8): Matrices and linear systems (cont.): independence and general solutions, Wronskian of solutions; general solutions of nonhomogeneous systems; solving an initial-value problem for a linear system of ODEs, augmented coefficient matrix of a linear system of algebraic equations, elementary row operations, an example [pages 295-300 of Section 5.1]
  Reading assignment: Nonhomogeneous solutions [pages 300-301 of Section 5.1]
  Homework: click on the link to download the assigned homework.
  FFT: Problems 5.1/43.
  The complete Homework 10 is due on November 15 (Friday).
• Lecture 35 (Mon, Nov 11): Matrices and linear systems (cont.): general solutions of homogeneous systems and of non-homogeneous systems [pages 300-301 of Sec. 5.1]
  The eigenvalue method for homogeneous systems: idea, definition of eigenvalues of a square matrix and corresponding eigenvectors, characteristic equation of a square matrix, eigenvalue solution of x'=Ax - the case of distinct real eigenvalues, examples [pages 304-308 of Sec. 5.2]
  Homework: Problems 5.2/5, 6, 17.
  FFT: Problems 5.2/27.
• Lecture 36 (Wed, Nov 13): The eigenvalue method for homogeneous systems (cont.): eigenvalue solution of x'=Ax - the case of distinct complex eigenvalues, examples [pages 311-315 of Sec. 5.2]
  Homework: Problems 5.2/11, 24.
• Lecture 37 (Fri, Nov 15): Matrix exponentials and linear systems: definition of an exponential of a square matrix, examples [pages 351-353 of Sec. 5.5]
  Reading assignment: definition of a nilpotent matrix, Examples 3 and 4 [pages 535-354 of Sec. 5.5]
  Homework: Problems 5.5/21, 25, 32 [hint for Problem 25: write A as a sum of a matrix D=2I and a nilpotent matrix B, then the fact that DB=BD guarantees that e^(D+B)t=e^Dte^Bt and solve the problem as in Example 4 on page 354; hint for Problem 32: obviously, each matrix A commutes with its inverse, A⁻¹]
• Lecture 38 (Mon, Nov 18): Exam 3 [on the material from Sec. 7.1-7.4, 7.6, 5.1 covered in Lectures 21-24, 26-34]
• Lecture 39 (Wed, Nov 20): Matrix exponentials and linear systems (cont.): proof that the solution of the IVP x'=Ax, x(0)=x₀, is x(t)=e^Atx₀; computing e^At for A being a diagonal matrix and for
  ;
  proof that (S⁻¹AS)^k=S⁻¹A^kS and that e^S−1AS=S⁻¹e^AS, using this to write D:=S⁻¹AS, then A=SDS⁻¹ and e^At=Se^DtS⁻¹; if all eigenvalues of A are distinct and real, the matrix S is made by stacking together the eigenvectors of A (written as column vectors).
  Homework: click on the link to download the assigned homework.
  The complete Homework 11 is due on November 25 (Monday).
• Lecture 40 (Fri, Nov 22): Matrix exponentials and linear systems (cont.): remarks on exponentiating big matrices - to exponentiate a matrix A, find a matrix S so that D:=S⁻¹AS becomes block-diagonal with the blocks on the diagonal having simple structure, then exponentiate D by exponentiating each block, and then go back to obtain e^At=Se^DtS⁻¹.
  Homework: click on the link to download the assigned homework.
• Lecture 41 (Mon, Nov 25): Vector spaces, inner product, functions as vectors, etc.: vector spaces (linear spaces); inner product vector spaces, orthogonal vectors; functions as vectors; basis and dimension of a vector space; the space of polynomials of degree no more than n is (n+1)-dimensional.
  Homework: click on the link to download the assigned homework.
• Lecture 42 (Mon, Dec 2): Vector spaces, inner product, functions as vectors, etc. (cont.): more on inner product vector spaces; inner product in function spaces, weight function w(x); periodic functions of period 2L form a vector space.
  Homework: No additional homework assigned.
  The complete Homework 12 is due on December 6 (Friday).
• Lecture 43 (Wed, Dec 4): applications of Fourier series for analysis of time-periodic (or approximately time-periodic) signals; amplitude modulation in radio transmission; transient (going to 0 as t→∞) and persistent (not going to 0 as t→∞) parts of the solution of a forced oscillator with dissipation (due to resistance in motion through fluid in mechanics or to losses in resistors in electric circuits) - for a non-homogeneous linear constant-coefficient ODE these two parts correspond respectively to the general solution of the associated homogeneous equation and to the particular solution of the non-homogeneous equation; using complex amplitudes to find the persistent part (up to a phase) by representing cos(ωt) as Re(e^iωt).
• Lecture 44 (Fri, Dec 6): class cancelled due to weather.
• Final exam: Monday, Dec 9, 8:00-10:00 a.m.

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

Homework: It is absolutely essential to solve the assigned homework problems! Homework assignments will be given regularly throughout the semester and will be posted on this web-site. The homework will be due at the start of class on the due date. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. Your lowest homework grade will be dropped. Your homework should have your name clearly written on it, and should be stapled. The problems should be written in the order they are given. No late homework will be accepted (unless you have a really compelling reason for turning it late)!

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. 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). You should come to class on time; if you miss a quiz because you came late, you won't be able to make up for it.

Useful links: the academic calendar, the class schedules.

Policy on W/I Grades : From Sept 3 (Tue) to Oct 25 (Fri), you can withdraw from the course with an automatic "W". Dropping after Oct 28 (Mon) requires a petition to the Dean. (Such petitions are not often granted. 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.) Please check the dates in the Academic Calendar!

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 details on the University's policies concerning academic integrity see the Student's Guide to Academic Integrity at the Academic Integrity web-site. For information on your rights to appeal charges of academic misconduct consult the Academic Misconduct Code. Students are also bound by the provisions of the OU Student Conduct Code.

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.

Good to know: the greek_alphabet, some useful notations.