Week 9: 26 Oct, 27 Oct, 28 Oct, 29 Oct, 30 Oct,

# 2250 Lecture Record Week 9 F2009

Last Modified: November 04, 2009, 21:19 MST.    Today: July 18, 2018, 12:35 MDT.

## 26 Oct: Basic Laplace Theory. Sections 10.1,10.2.

  Drill:
Sampling in partial fractions.
Method of atoms in partial fractions.
Heaviside's coverup method.
Lecture: Basic Laplace theory.
Read 5.5, 5.6, ch6, ch7, ch8, ch9 later.
Direct Laplace transform == Laplace integral.
Def: Direct Laplace
transform == Laplace integral
== int(f(t)exp(-st),t=0..infinity)
== L(f(t)).
Introduction to Laplace's method.
The method of quadrature for higher order equations and systems.
Calculus for chapter one quadrature versus the Laplace calculus.
The Laplace integrator dx=exp(-st)dt.
The Laplace integral abbreviation L(f(t)) for the Laplace integral of f(t).
Lerch's cancelation law and the fundamental theorem of calculus.
A Brief Laplace Table
1, t, t^2, t^n, exp(at), cos(bt), sin(bt)


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)
Lecture: Delayed topics
Common errors in solving higher order equations.
Wrong number of atoms, duplicate atoms
Complex unit i appears in the cosine and sine factors
Inefficient analysis of atoms for complex conjugate root
Identifying atoms in linear combinations.
How to solve for c1, c2, etc when initial conditions are given.


## 27 Oct: Laplace theorems. Sections 10.2,10.3,10.4.

 Solution to 4.7-10: Subspace Criterion. Blackboard only.
Last day to submit chapter 1 extra credit problems.
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)



## 28 Oct: Applications of Laplace's method from 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
Damped oscillations
overdamped, critically damped, underdamped [Chapter 5]
phase-amplitude form of the solution [chapter 5]


## 29 Oct: Fusi and Richins

Exam 2 review, exam Problems 2,3,4. Exercises 10.1, 10.2, 10.3.

## 30 Oct: Mechanical oscillators. Resonance. Beats.

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.