2250-1 7:30am Lecture Record Week 10 S2010

Last Modified: March 29, 2010, 04:43 MDT.    Today: October 16, 2017, 16:16 MDT.

Week 10, Mar 15 to 19: Sections 10.3, 10.4, 10.5, EPbvp7.6

15-16 Mar: Applications of Laplace's method from 10.3, 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

17 Mar: Mechanical oscillators. Resonance. Beats. Sections 10.4, 10.5

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.

17-18 Mar: Murphy

Exam 2 at 6:45am to 8:30am in JTB 140 on 17 Mar, 12:45-3:00pm in WEB 104 on 18 Mar.

19 Mar: Piecewise Functions. Shifting. Section 10.5 and EPbvp supplement 7.6.

 Piecewise Functions
   Unit Step
   Pulse
   Ramp
 Second shifting Theorems
   e^{-as}L(f(t))=L(f(t-a)step(t-a))
   L(g(t)step(t-a))=e^{-as}L(g(t+a))
 Piecewise defined periodic waves
   Square wave
   Triangular wave
   Sawtooth
   Rectified sine
   Half-wave rectified sine
   Parabolic wave
 Periodic function theorem
    Laplace of the square wave, tanh function.
    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 (85.7 K, pdf, 01 Jan 2010)
    Manuscript: DE systems, examples, theory (785.8 K, pdf, 16 Nov 2008)
    Slides: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)
    Slides: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)
    Slides: Home heating, attic, main floor, basement (73.8 K, pdf, 30 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: 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 (8.9 K, txt, 31 Dec 2009)
    Text: Final exam study guide (8.1 K, txt, 30 Dec 2009)