# 2250-4 12:55pm Lecture Record Week 10 S2010

Last Modified: March 25, 2010, 17:51 MDT.    Today: September 24, 2018, 02:05 MDT.

## 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


## 16 Mar: Mechanical oscillators. Resonance. Beats. Sections 10.4, 10.5

Basic Theorems of Laplace Theory
Periodic function theorem
Proof
Convolution theorem application
L(cos t)L(sin t) = L(0.5 t sin(t))
Applications
How to solve differential equations

Specialized models.
Pure Resonance x''+x=cos(t)
Solution explosion, unbounded solution x=(1/2)t \sin t.


## 18 Mar: Periodic piecewise functions. Second shifting theorem.

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. Some engineering functions unit step ramp sawtooth wave 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)) LRC Circuit 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.

## 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.