Edwards-Penney, sections 5.1, 5.2, 5.3, 5.4 The textbook topics, definitions and theorems

Edwards-Penney 5.1, 5.2, 5.3, 5.4 (15.6 K, txt, 19 Dec 2013)

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. 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. 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); PROBLEM 4.7-26. Done last time. How to 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, 26 Feb 2013)Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012) PICARD'S THEOREM. It says that nth order equations have a solution space of dimension n. EULER'S THEOREM. 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. 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. 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, {1, exp(-10x)} is a basis for the solution space of the differential equation. Then the general solution is y = c1 (1) + c2 (exp(-10x)).Slides## Survey of topics for this week.

Linear DE Slides.: Picard-Lindelof, linear nth order DE, superposition (181.5 K, pdf, 03 Mar 2012)Slides: How to solve linear DE or any order (153.3 K, pdf, 25 Feb 2014)Slides: Atoms, Euler's theorem, 7 examples (130.5 K, pdf, 26 Feb 2013)Slides

Theory of Higher Order Constant Equations:Homogeneous and non-homogeneous structure. Superposition. Picard's Theorem. Solutions form a vector space of dimension n Dimension of the solution set = n = order of highest derivative Euler Solution Atoms. Definition of atom. Independence of atoms. Euler's theorem. Real roots Non-real roots [complex roots]. How to deal with conjugate pairs of factors (r-a-ib), (r-a+ib). The Euler formula exp(i theta)=cos(theta) + i sin(theta). How to solve homogeneous equations: Use Euler's theorem to find a list of n distinct solution atoms. Examples: y''=0, y''+3y'+2y=0, y''+y'=0, y'''+y'=0.Second 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.

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.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.

Continued: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. 5.1-48: 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.

- References: Sections 5.4, 5.6. Forced oscillations.

Lab 8. Details forthcoming in Week 8. Lab 7 problems are due Thursday.

Lecture: Applications. Damped and undamped motion.Last time: Theory of equations and 5.3-32. Spring-mass equation, Spring-mass DE derivation Spring-mass equation mx''+cx'+kx=0 and its physical parameters. Spring-mass system, my'' + cy' + ky = 0 harmonic oscillation, y'' + omega^2 y = 0, Forced systems. Forcing terms in mechanical systems. Speed bumps. 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.: Unforced vibrations 2008 (647.6 K, pdf, 28 Feb 2014)Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012)Slides: Forced damped vibrations (264.0 K, pdf, 08 Mar 2014)Slides: Forced vibrations and resonance (253.0 K, pdf, 08 Mar 2014)Slides: Resonance and undetermined coefficients, cafe door, pet door, phase-amplitude (178.0 K, pdf, 08 Mar 2014)Slides: Electrical circuits (112.9 K, pdf, 08 Mar 2014)Slides