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.Drilltop=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.

- References: Sections 5.4, 5.6. Forced oscillations.

Problem sessionAll problems 4.6-4.7. Theory of equations and 5.3-32. All of 5.2, 5.3 discussed.

Slides on Section 5.4Damped 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.

More exam 2 review, problems 4,5Partly 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 5.4-20The 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) Plots can be made from these answers directly. 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)/2Lecture: 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 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