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

Last Modified: March 26, 2012, 04:42 MDT.    Today: December 15, 2017, 10:44 MST.

## 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.
```

## 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]
```