Google Search in:
Week 9: 26 Oct,  27 Oct,  28 Oct,  29 Oct,  30 Oct,

2250 Lecture Record Week 9 F2009

Last Modified: November 04, 2009, 21:19 MST.    Today: December 17, 2017, 20:10 MST.

Week 9, Oct 26 to 30: Sections 10.1,10.2,10.3,10.4,10.5

26 Oct: Basic Laplace Theory. Sections 10.1,10.2.

  Drill:
   Sampling in partial fractions.
   Method of atoms in partial fractions.
   Heaviside's coverup method.
Lecture: Basic Laplace theory.
 Reading: Chapter 10.
          Read 5.5, 5.6, ch6, ch7, ch8, ch9 later.
 Direct Laplace transform == Laplace integral.
 Def: Direct Laplace
      transform == Laplace integral
                == int(f(t)exp(-st),t=0..infinity)
                == L(f(t)).
 Introduction to Laplace's method.
   The method of quadrature for higher order equations and systems.
   Calculus for chapter one quadrature versus the Laplace calculus.
   The Laplace integrator dx=exp(-st)dt.
   The Laplace integral abbreviation L(f(t)) for the Laplace integral of f(t).
   Lerch's cancelation law and the fundamental theorem of calculus.
 A Brief Laplace Table
   1, t, t^2, t^n, exp(at), cos(bt), sin(bt)

Slides: Laplace and Newton calculus. Photos. (145.3 K, pdf, 01 Nov 2009)
Slides: Intro to Laplace theory. Calculus assumed. (109.5 K, pdf, 01 Nov 2009)
Slides: Laplace rules (112.2 K, pdf, 01 Nov 2009)
Slides: Laplace table proofs (130.3 K, pdf, 01 Nov 2009)
Lecture: Delayed topics
  Common errors in solving higher order equations.
    Wrong number of atoms, duplicate atoms
    Complex unit i appears in the cosine and sine factors
    Inefficient analysis of atoms for complex conjugate root
  Identifying atoms in linear combinations.
  How to solve for c1, c2, etc when initial conditions are given.

27 Oct: Laplace theorems. Sections 10.2,10.3,10.4.

 Solution to 4.7-10: Subspace Criterion. Blackboard only.
 Last day to submit chapter 1 extra credit problems.
 A brief Laplace table.
   Forward table.
   Backward table.
   Extensions of the Table.
 Laplace rules.
   Linearity.
   The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
   Shift theorem.
   Parts theorem.
 Finding Laplace integrals using Laplace calculus.
 Solving differential equations by Laplace's method.
 Basic Theorems of Laplace Theory
   Lerch's theorem
   Linearity.
   The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
   Shift theorem L(exp(at)f(t)) = L(f(t))|s->(s-a)
   Parts theorem L(y')=sL(y)-y(0)

 

28 Oct: Applications of Laplace's method from 10.4, 10.5.

 Solving y' = -1, y(0)=2
 Solving y''+y=0, y(0)=0, y'(0)=1
 Solving y''+y=1, y(0)=y'(0)=0
 Solving y''+y=cos(t), y(0)=y'(0)=0
 Computing Laplace integrals
 Solving an equation L(y(t))=expression in s for y(t)
 Dealing with complex roots and quadratic factors
 Partial fraction methods
 Using the s-differentiation theorem
 Using the shifting theorem
 Harmonic oscillator y''+a^2 y=0
 Damped oscillations
   overdamped, critically damped, underdamped [Chapter 5]
   phase-amplitude form of the solution [chapter 5]

29 Oct: Fusi and Richins

Exam 2 review, exam Problems 2,3,4. Exercises 10.1, 10.2, 10.3.

30 Oct: Mechanical oscillators. Resonance. Beats.

Basic Theorems of Laplace Theory
   Periodic function theorem
     Proof
     Some engineering functions
     unit step
     ramp
     sawtooth wave
     other periodic waves next Monday
   Convolution theorem application
     L(cos t)L(sin t) = L(0.5 t sin(t))
 Applications
   How to solve differential equations
   LRC Circuit
   Second shifting rule
   Specialized models.
     Pure Resonance x''+x=cos(t)
       Solution explosion, unbounded solution x=(1/2)t \sin t.
       Resonance examples: Soldiers marching in cadence, Tacoma narrows bridge,
              Wine Glass Experiment. Theodore Von Karman and vortex shedding. 
              Cable model of the Tacoma bridge, year 2000. Resonance explanations.
     Beats x''+x=cos(2t)
        Graphics for beats [x=sin(10 t)sin(t/2)], slowly-oscillating envelope,
        rapidly oscillating harmonic with time-varying amplitude.


    Laplace theory references
    Slides: Laplace and Newton calculus. Photos. (145.3 K, pdf, 01 Nov 2009)
    Slides: Intro to Laplace theory. Calculus assumed. (109.5 K, pdf, 01 Nov 2009)
    Slides: Laplace rules (112.2 K, pdf, 01 Nov 2009)
    Slides: Laplace table proofs (130.3 K, pdf, 01 Nov 2009)
    Slides: Laplace examples (101.2 K, pdf, 07 Nov 2009)
    Slides: Piecewise functions and Laplace theory (64.7 K, pdf, 01 Nov 2009)
    MAPLE: Maple Lab 7. Laplace applications (84.3 K, pdf, 19 Jul 2009)
    Slides: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)
    Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)
    Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)
    Manuscript: Heaviside's method 2008 (186.8 K, pdf, 20 Oct 2009)
    Manuscript: DE systems, examples, theory (785.8 K, pdf, 16 Nov 2008)
    Manuscript: Laplace theory 2008 (350.5 K, pdf, 06 Mar 2009)
    Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1968.3 K, pdf, 13 Nov 2003)
    Text: Laplace theory problem notes F2008 (8.1 K, txt, 21 Nov 2008)
    Text: Final exam study guide (7.6 K, txt, 12 Dec 2009)
    Slides: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)