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

Drill: Sampling in partial fractions. Method of atoms in partial fractions. Heaviside's coverup method.Lecture: Basic Laplace theory.Reading: Chapter 10. 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)

Lecture: Delayed topicsCommon 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.

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 TheoryLerch'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)

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]

Basic Theorems of Laplace TheoryPeriodic 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))ApplicationsHow 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.