TopicsSections 3.1 to 3.6 The textbook topics, definitions, examples and theorems

Edwards-Penney Ch 3, 3.1 to 3.4 (16.5 K, txt, 04 Jan 2015)

Edwards-Penney Ch 3, 3.5 to 3.7 (17.7 K, txt, 02 Jan 2015)

Linear Differential Equations of Order nPICARD FAILURE. Although Picard's structure theorem does not provide an algorithm for construction of n independent solutions, the theorems of Euler do that. Combined, there is an easy path to finding a basis for the solution space of an nth order linear differential equation. EULER'S THEOREM for CONSTRUCTING a SOLUTION BASIS It says y=exp(rx) is a solution of ay'' + by' + cy = 0 <==> r is a root of the characteristic equation ar^2+br+c=0. REAL EXPONENTIALS: If the root r is real, then the exponential is a real solution. THEOREM. A real root r=a (positive, negative or zero) produces one Euler solution atom exp(ax). COMPLEX EXPONENTIALS: If a nonreal root r=a+ib occurs, a complex number, then there is a conjugate root a-ib. The pair of roots produce two real solutions from EULER'S FORMULA (a trig topic): exp(i theta) = cos(theta) + i sin(theta) Details to obtain the two solutions will be delayed. The answer is THEOREM. A conjugate root pair a+ib,a-ib produces two independent Euler solution atoms exp(ax) cos(bx), exp(ax) sin(bx). HIGHER MULTIPLICITY For roots of the characteristic equation of multiplicity greater than one, there is a correction to the answer obtained in the two theorems above: Multiply the answers from the theorems by powers of x until the number of Euler solution atoms produced equals the multiplicity. EXAMPLE: If r=3,3,3,3,3 (multiplicity 5), then multiply exp(3x) by 1, x, x^2, x^3, x^4 to obtain 5 Euler solution atoms. EXAMPLE: If r=5+3i,5+3i (multiplicity 2), then there are roots r=5-3i,5-3i, making 4 roots. Multiply the two Euler atoms exp(5x)cos(3x), exp(4x)sin(3x) by 1, x to obtain 4 Euler solution atoms. SHORTCUT: The characteristic equation can be synthetically formed from the differential equation ay''+by'+cy=0 by the formal replacement y ==> 1, y' ==> r, y'' ==> r^2. EULER'S SOLUTION ATOMS Leonhard Euler described a complete solution to finding n independent solutions in the special case when the coefficients are constant. The Euler solutions are calledatomsin these lectures.Slides: Shock-less auto. Rolling wheel on a spring. Swinging rod. Mechanical watch. Bike trailer. Physical pendulum.Chapter 3 references: Resonance and undetermined coefficients, cafe door, pet door, phase-amplitude (0.0 K, pdf, 31 Dec 1969)Slides: Electrical circuits (0.0 K, pdf, 31 Dec 1969)Slides: Unforced vibrations 2008 (0.0 K, pdf, 31 Dec 1969)Slides: Forced damped vibrations (0.0 K, pdf, 31 Dec 1969)Slides: Forced vibrations and resonance, Millenium Bridge, Wine Glass, Tacoma Narrows (0.0 K, pdf, 31 Dec 1969)Slides: Forced undamped vibrations (0.0 K, pdf, 31 Dec 1969)Slides: phase-amplitude, cafe door, pet door, damping classification (0.0 K, pdf, 31 Dec 1969) Projection: glass-breaking video. Wine glass experiment. Tacoma narrows.Slides: Wine glass breakage (avi) (0.0 K, avi, 31 Dec 1969)Video: Glass breakage in slow motion, MIT (0.0 K, 31 Dec 1969)2015 Video: Glass breakage in slow motion, MIT (same video) (0.0 K, 31 Dec 1969)2009 Video: Same 2009 Glass Breakage, local copy (0.0 K, mp4, 31 Dec 1969)Video: Wine glass experiment (12mb mpg, 2min) (0.0 K, mpg, 31 Dec 1969)Video: Tacoma Narrows Bridge Nov 7, 1940 (18mb mpg, 4min) (0.0 K, mpg, 31 Dec 1969) THE TERM ATOM. The termVideoatomabbreviatesEuler solution atom of a linear differential equation. The main theorem says that the answer to a homogeneous constant coefficient linear differential equation of higher order is a linear combinations of atoms. EULER SOLUTION ATOMS for LINEAR DIFFERENTIAL EQUATIONS DEF. Base atoms are 1, exp(a x), cos(b x), sin(b x), exp(ax)cos(bx), exp(ax)sin(bx). DEF: atom = x^n (base atom) for n=0,1,2,... THEOREM. Euler solution atoms are independent. THEOREM. Solutions of constant-coefficient homogeneous differential equations are linear combinations of a complete set of Euler solution atoms. EXAMPLE. The equation y''+10y'=0 has characteristic equation r^2+10r=0 with roots r=0, r=-10. Then Euler's theorem says exp(0x) and exp(-10x) are solutions. By vector space dimension theory, the Euler solution atoms 1, exp(-10x) are a basis for the solution space of the differential equation. Then the general solution is y = a linear combination of the Euler solution atoms y = c1 (1) + c2 (exp(-10x)).SOLUTION ATOMS and INDEPENDENCE. Def. atom=x^n(base atom), n=0,1,2,3,... where for a nonzero real and b>0, base atom = 1, cos(bx), sin(bx), exp(ax), exp(ax) cos(bx), exp(ax) sin(bx) "atom" abbreviates "Euler solution atom of a linear differential equation" THEOREM. Euler Solution Atoms are Independent. EXAMPLE. Show 1, x^2, x^9 are independent [atom theorem] EXAMPLE. Show 1, x^2, x^(3/2) are independent [Wronskian test] PARTIAL FRACTION THEORY REVIEW. MAPLE ASSIST. top:=x-1; bottom:=(x+1)*(x^2+1); convert(top/bottom,parfrac,x); top:=x-1; bottom:=(x+1)^2*(x^2+1)^2; convert(top/bottom,parfrac,x); EXAMPLE. Solve y''+10y'=0 for general solution y=c1 + c2 exp(-10x) TOOLKIT for SOLVING LINEAR CONSTANT DIFFERENTIAL EQUATIONS Picard: Order n of a DE = dimension of the solution space. General solution = linear combination n independent atoms. Euler's theorem(s), an algorithm for finding solution atoms.Summary for Higher Order Differential Equations: Atoms, Euler's theorem, 7 examples (130.5 K, pdf, 25 Feb 2013)Slides: Base atom, atom, basis for linear DE (106.8 K, pdf, 07 Feb 2018)SlidesSecond order equations.Homogeneous equation. Harmonic oscillator example y'' + y=0. Picard-Lindelof theorem. Dimension of the solution space. Structure of solutions. Non-homogeneous equation. Forcing term.Nth order equations.Solution space theorem for linear differential equations. Superposition. Independence and Wronskians. Independence of atoms. Main theorem on constant-coefficient equations THEOREM. Solutions are linear combinations of atoms. Euler's substitution y=exp(rx). Shortcut to finding the characteristic equation. Euler's basic theorem: y=exp(rx) is a solution <==> r is a root of the characteristic equation. Euler's multiplicity theorem: y=x^n exp(rx) is a solution <==> r is a root of multiplicity n+1 of the characteristic equation. How 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.Constant coefficient equations with complex roots.Applying Euler's theorems to solve a DE. Examples of order 2,3,4. Exercises 3.1, 3.2, 3.3. 3.1-34: y'' + 2y' - 15y = 0 3.1-36: 2y'' + 3y' = 0 3.1-38: 4y'' + 8y' + 3y = 0 3.1-40: 9y'' -12y' + 4y = 0 3.1-42: 35y'' - y' - 12y = 0 3.1-46: Find char equation for y = c1 exp(10x) + c2 exp(100x) 3.1-48: 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). 3.2-18: 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. 3.2-22: 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). 3.3-8: y'' - 6y' + 13y = 0 (r-3)^2 +4 = 0 3.3-10: 5y'''' + 3y''' = 0 r^3(5r+3) = 0 3.3-16: y'''' + 18y'' + 81 y = 0 (r^2+9)(r^2+9) = 0 Check all answers with Maple/MuPad, using this example: de:=diff(y(x),x,x,x,x)+18*diff(y(x),x,x)+81*y(x) = 0; # Maple/MuPad dsolve(de,y(x)); # maple only ode::solve(de,y(x)); # MuPad only 3.3-32: 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/MuPad, 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.Thursday Exam Review: Intro to problems 1,2,3,4,5Presentation of Problem 5 from the sample exam. Blackboard photos published at the course web site.Slides: Shock-less auto. Rolling wheel on a spring. Swinging rod. Mechanical watch. Bike trailer. Physical pendulum.Chapter 3 references. Sections 3.4, 3.5, 3.6. Forced/Unforced oscillations.: Resonance and undetermined coefficients, cafe door, pet door, phase-amplitude (178.0 K, pdf, 08 Mar 2014)Slides: Electrical circuits (112.8 K, pdf, 19 Feb 2016)Slides: Unforced vibrations 2008 (647.6 K, pdf, 27 Feb 2014)Slides: Forced damped vibrations (263.9 K, pdf, 10 Feb 2016)Slides: Forced vibrations and resonance, Millenium Bridge, Wine Glass, Tacoma Narrows (253.0 K, pdf, 08 Mar 2014)Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012)Slides: phase-amplitude, cafe door, pet door, damping classification (136.0 K, pdf, 08 Mar 2014) Projection: glass-breaking video. Wine glass experiment. Tacoma narrows.Slides: Wine glass breakage (avi) (260.5 K, avi, 18 Feb 2015) 2015 Video: Glass breakage in slow motion, MIT 2009 Video: Glass breakage in slow motion, MIT (same video)Video: Same 2009 Glass Breakage, local copy (12992.3 K, mp4, 16 Feb 2016)Video: Wine glass experiment (12mb mpg, 2min) (12493.8 K, mpg, 01 Apr 2008)Video: Tacoma Narrows Bridge Nov 7, 1940 (18mb mpg, 4min) (18185.8 K, mpg, 01 Apr 2008)Video