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2250-4 12:55pm Lecture Record Week 10 S2010

Last Modified: March 25, 2010, 17:51 MDT.    Today: October 19, 2017, 10:34 MDT.

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

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

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

Basic Theorems of Laplace Theory
   Periodic function theorem
     Proof
   Convolution theorem application
     L(cos t)L(sin t) = L(0.5 t sin(t))
 Applications
   How to solve differential equations

   Specialized models.
     Pure Resonance x''+x=cos(t)
       Solution explosion, unbounded solution x=(1/2)t \sin t.

18 Mar: Periodic piecewise functions. Second shifting theorem.

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. Some engineering functions unit step ramp sawtooth wave 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)) LRC Circuit 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.

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.
    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)