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

Last Modified: March 29, 2010, 04:43 MDT.    Today: September 24, 2018, 01:27 MDT.

## 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.
```