PROBLEM SESSIONChapter 4 exercises. Next package is 4.5-6,24,28; 4.6-2; 4.7-10,20,26. 4.5: Discuss 4.5-6[pivot thm], 4.5-24[pivot thm], 4.5-28[proof]. 4.6: Discuss 4.6-2 solution details. 4.7: Already discussed: 4.7-20,26. We'll discuss 4.7-10. You are expected to read 4.7-1 to 4.7-12 and report "yes a subspace" or "not a subspace" the whole task done in less than 5 minutes. Answers in B.O.B. Details of proof will take 3-10 minutes per problem. 4.7 Independence. Classify as indep/dep. For indep, supply a reason, from these theorems: THM. Atoms are indep; THM. Wonskian nonzero => indep. 4.7-13: sin x, cos x 4.7-14: exp(x), x exp(x) 4.7-15: 1+x,1-x, 1-x^2 4.7-16: 1+x, x+x^2, 1-x^2 4.7-17: cos 2x, sin^2 x, cos^2 x 4.7-18: 2 cos x + 3 sin x, 4 cos x + 5 sin x

REVIEW 5.1, 5.2, 5.3How to solve any constant-coefficient nth order homogeneous differential equation. 1. Find the n roots of the characteristic equation. 2. Apply Euler's theorems to find n distinct solution atoms. 2a. Find the base atom for each distinct real root. Multiply each base atom by powers 1,x,x^2, ... until the number of atoms created equals the root multiplicity. 2b. Find the pair of base atoms for each conjugate pair of complex roots. Multiply each base atom by powers 1,x,x^2, ... until the number of atoms created equals the root multiplicity. 3. Report the general solution as a linear combination of the n atoms. Picard's Theorem for higher order DE and systems. Solution space dimension.Constant coefficient equations with complex roots.Applying Euler's theorems to solve a DE. Examples of order 2,3,4. Exercises 5.1, 5.2, 5.3. 5.1-34: y'' + 2y' - 15y = 0 5.1-36: 2y'' + 3y' = 0 5.1-38: 4y'' + 8y' + 3y = 0 5.1-40: 9y'' -12y' + 4y = 0 5.1-42: 35y'' - y' - 12y = 0 5.1-46: Find char equation for y = c1 exp(10x) + c2 exp(100x) 5.1-48: Find char equation for y = l.c. of atoms exp(r1 x), exp(r2 x) where r1=1+sqrt(2) and r2=1-sqrt(2). 5.2-18: Solve for c1,c2,c3 given initial conditions and general solution. y(0)=1, y'(0)=0, y''(0)=0 y = c1 exp(x) + c2 exp(x) cos x + c3 exp(x) sin x. 5.2-22: Solve for c1 and c2 given initial conditions y(0)=0, y'(0)=10 and y = y_p + y_h = -3 + c1 exp(2x) + c2 exp(-2x). 5.3-8: y'' - 6y' + 13y = 0 (r-3)^2 +4 = 0 5.3-10: 5y'''' + 3y''' = 0 r^3(5r+3) = 0 5.3-16: y'''' + 18y'' + 81 y = 0 (r^2+9)(r^2+9) = 0 Check all answers with Maple, using this example: de:=diff(y(x),x,x,x,x)+18*diff(y(x),x,x)+81*y(x) = 0; dsolve(de,y(x)); 5.3-32: Theory of equations and Euler's method. Char equation is r^4 + r^3 - 3r^2 -5r -2 = 0. Use the rational root theorem and long division to find the factorization (r+1)^3(r-2)=0. Check the root answer in Maple, using the code solve(r^4 + r^3 - 3*r^2 -5*r -2 = 0,r); The answer is a linear combination of 4 atoms, obtained from the roots -1,-1,-1,2.

Second order and higher order differential Equations.Picard theorem for second order equations, superposition, solution space structure, dimension of the solution set. Euler's theorem. Quadratic equations again. Constant-coefficient second order homogeneous differential equations. Characteristic equation and its factors determine the atoms. Applications. Spring-mass system, RLC circuit equation. harmonic oscillation, Sample equations: y''=0, y''+2y'+y=0, y''-4y'+4y=0, y''+y=0, y''+3y'+2y=0, mx''+cx'+kx=0, LQ''+RQ'+Q/C=0. How to solve examples like the 5.1,5.2,5.3 problems. Solving a DE when the characteristic equation has complex roots. Higher order equations or order 3 and 4. Finding 2 atoms from one pair of complex conjugate roots. Why the complex conjugate root identifies the same two atoms. Equations with both real roots and complex roots. An equation with 4 complex roots. How to find the 4 atoms.

- References: Sections 5.4, 5.6. Forced oscillations.

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. Span theorem. Blackboard only.

- References: Sections 5.4, 5.6. Forced oscillations.

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.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)/2Partly 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.## Basic Laplace Theory. Section 10.1

More exam 2 review, problems 4,5Lecture: 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