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2250-1 7:30am Lecture Record Week 10 S2012

Last Modified: March 26, 2012, 04:42 MDT.    Today: December 15, 2017, 10:44 MST.

Week 10, Mar 19 to 23: Sections 10.2, 10.3, 10.4, 10.5, EPbvp7.6

Mar 22: Patrick Bardsley

Exam 2, problems 1,2,3 at 7:30am to 9:25am in JTB 140 on Thursday, March 22. Study the S2010 and F2010 exam solution keys for similar problems.

19 Mar: Tables, Rules of Laplace's Calculus 10.2, 10.3.

 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)

 

20 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

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

Basic Theorems of Laplace Theory
     Some engineering functions
     unit step
     ramp
     sawtooth wave
     other periodic waves next Monday
 Applications
   How to solve differential equations
   LRC Circuit
   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.
    Oscillations. Mechanical and Electrical.
    Slides: Electrical circuits (124.8 K, pdf, 04 Mar 2012)
    Slides: Forced damped vibrations (280.7 K, pdf, 04 Mar 2012)
    Slides: Forced vibrations and resonance (240.3 K, pdf, 04 Mar 2012)
    Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012)
    Slides: Resonance and undetermined coefficients (199.4 K, pdf, 03 Mar 2012)
    Slides: Unforced vibrations 2008 (667.9 K, pdf, 03 Mar 2012)

23 Mar: Piecewise Functions. Section 10.5 and EPbvp supplement 7.6.

Piecewise Functions
   Unit Step: step(t)=1 for t>=0, step(t)=0 for t<0.
   Pulse: pulse(t,a,b)=step(t-a)-step(t-b)
   Ramp: ramp(t-a)=(t-a)step(t-a)
   L(step(t-a)) = (1/s) exp(-as) [for a >= 0 only]
Piecewise defined periodic waves
   Square wave
   Triangular wave
   Sawtooth
Integral Theorem
   L(int(g(x),x=0..t)) = s L(g(t))
   Applications to computing ramp(t-a)
    L(ramp(t-a)) = (1/s^2) exp(-as) [for a >= 0 only]
    Laplace theory references
    Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 04 Mar 2012)
    Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 19 Mar 2012)
    Slides: Laplace rules (160.3 K, pdf, 04 Mar 2012)
    Slides: Laplace table proofs (169.6 K, pdf, 04 Mar 2012)
    Slides: Laplace examples (149.1 K, pdf, 04 Mar 2012)
    Slides: Piecewise functions and Laplace theory (108.1 K, pdf, 04 Mar 2012)
    MAPLE: Maple Lab 7. Laplace applications (60.6 K, pdf, 09 Dec 2011)
    Manuscript: DE systems, examples, theory (785.8 K, pdf, 16 Nov 2008)
    Slides: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012)
    Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012)
    Slides: Home heating, attic, main floor, basement (109.8 K, pdf, 04 Mar 2012)
    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 (1068.7 K, pdf, 28 Nov 2010)
    Text: Laplace theory problem notes F2008 (8.9 K, txt, 18 Nov 2010)
    Text: Final exam study guide (8.2 K, txt, 11 Dec 2011)