# 2250-2 7:30am Lecture Record Week 9 F2010

Last Modified: October 31, 2010, 08:28 MDT.    Today: July 18, 2018, 12:33 MDT.

## 25, 27, 28 Oct Michal and Laura: Midterm Exam 2, problems 1,2,3

Attend any one exam session. JWB 335 at 12:50pm on Monday 25 Oct or Wed 27 Oct. WEB 103 at 7:25am on Thursday 28 Oct. The best sample exams are S2010 and F2009.

## 25 Oct: Damped and Undamped Motion. Section 5.4

```Lecture: Applications. Damped and undamped motion.
Last time: Theory of equations and 5.3-32.
Problems discussed in class: 5.3-10,20,26
Spring-mass equation,
LRC-circuit equation,
Spring-mass DE and RLC-circuit DE derivations.
Electrical-mechanical analogy.
The RLC circuit equation and its physical parameters.
Spring-mass equation mx''+cx'+kx=0 and its physical parameters.
Forced systems.
Forcing terms in mechanical systems. Speed bumps.
Forcing terms in electrical systems. Battery. Generator.
Harmonic oscillations: sine and cosine terms of frequency omega.
Damped and undamped equations. Phase-amplitude form.
Slides:
Shock-less auto.
Rolling wheel on a spring.
Swinging rod.
Mechanical watch.
Bike trailer.
Physical pendulum.
Solving more complicated homogeneous equations.
Example: Linear DE given by roots of the characteristic equation.
Example: Linear DE given by factors of the characteristic polynomial.
Example: Construct a linear DE of order 2 from a list of two atoms that must be solutions.
Example: Construct a linear DE from roots of the characteristic equation.
Example: Construct a linear DE from its general solution.
Drill
top=x-1, bot=(x+1)(x^2+4)
top/bot = A/(x+1)+(Bx+C)/(x^2+4); find A,B,C.
Sampling in partial fractions.
Method of atoms in partial fractions.
Heaviside's coverup method.
Solution to 4.7-10: Subspace Criterion. Blackboard only.
```

## 26 Oct: Applications x'' + px' + qx=0. Sections 5.4.

```Problem session
All problems 4.6-4.7.
Theory of equations and 5.3-32.
All of 5.2, 5.3 discussed.
```
```Slides on Section 5.4
Damped oscillations
overdamped, critically damped, underdamped [Chapter 5]
phase-amplitude form of the solution [chapter 5]
Undamped oscillations.
Harmonic oscillator.
Partly solved 5.4-20.
See the FAQ at the web site for answers and details.
Beats.
Decomposition of x(t) into two harmonic oscillations of different
natural frequencies. Envelope curves. Sound waves.
Pure resonance.
Pendulum.
Cafe door.
Pet door.
Over-damped, Critically-damped and Under-damped behavior.
pseudoperiod.
Washing machine.
```

## 27 Oct: Basic Laplace Theory. Section 10.1

Exam Review S2010 exam 2 details, problems 4,5 Problems chapter 5 Examples and solutions Lecture: Introduction to Laplace theory. Newton and Laplace: portraits of the Two Greats [slides]. Method of quadrature. Comparison of Newton calculus and Laplace calculus. Laplace integral. The Laplace integrator dx=exp(-st)dt. Direct Laplace transform == Laplace integral.

## 29 Oct: Intro to Laplace Theory. Sections 10.1,10.2,10.3.

```More exam 2 review, problems 4,5
Partly solved 5.4-34.
The DE is 3.125 x'' + cx' + kx=0. The characteristic equation
is 3.125r^2 + cr + kr=0 which factors into 3.125(r-a-ib)(r-a+ib)=0
having complex roots a+ib, a-ib. Problems 32, 33 find the numbers
a, b from the given information. This is an inverse problem, one
in which experimental data is used to discover the differential
equation model. The book uses its own notation for the symbols
a,b: a ==> -p and b ==> omega1. Because the two roots a+ib, a-ib
determine the quadratic equation, then c and k are known in terms
of symbols a,b. See also the web site FAQ for more details.
Partly solved 5.4-20
The problem breaks into two distinct initial value problems:
(1)   2x'' + 16x' + 40x=0, x(0)=5, x'(0)=4
Characteristic equation  2(r^2+8r+20)=0. Roots r=-4+2i,r=-4-2i.
Solution Atoms=e^{-4t}cos 2t, e^{-4t}sin 2t. Underdamped.

(2)   2x'' + 0x' + 40x=0, x(0)=5, x'(0)=4
Characteristic equation 2(r^2+0+20)=0. Roots r=sqrt(20)i,r=-sqrt(20)i.
Solution Atoms=cos( sqrt(20)t), sin( sqrt(20)t).
Each system has general solution a linear combination of the solution atoms.
Evaluate the constants in the linear combination, in each of the two
cases, using the initial conditions x(0)=5, x'(0)=4. There are two linear algebra
problems to solve.
Answers: (1)  Coefficients 5, 2  for 2x'' + 16x' + 40x=0
Amplitude sqrt(5^2 + 12^2) = 13
(2)  Coefficients 5, 2/sqrt(5) for 2x'' + 0x' + 40x=0
Amplitude sqrt(5^2 + 4/5) = sqrt(129/5)
Write each solution in phase-amplitude form, a trig problem. See section
5.4 for specific instructions. The book's answers:
(1) tan(alpha) = 5/12   (2) tan(alpha) = 5 sqrt(5)/2
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
Photos of Laplace and Newton: slides
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)
Some Laplace rules: Linearity, Lerch
Laplace's L-notation and the forward table
```

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)