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

**Last Modified**: March 29, 2010, 04:43 MDT. **Today**: February 25, 2018, 00:59 MST.
### 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)