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

**Last Modified**: March 26, 2012, 04:42 MDT. **Today**: March 24, 2018, 20:53 MDT.
### 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)