## 2280 12:55pm Lectures Week 5 S2018

Last Modified: February 13, 2018, 12:38 MST.    Today: December 10, 2018, 08:58 MST.

### Monday-Tuesday: Ch 3

```Topics
Sections 3.1 to 3.6
The textbook topics, definitions, examples and theoremsEdwards-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 n
PICARD 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 called atoms in these
lectures.Slides:
Shock-less auto.
Rolling wheel on a spring.
Swinging rod.
Mechanical watch.
Bike trailer.
Physical pendulum.

Chapter 3 referencesSlides: 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.
Video: Wine glass breakage (avi) (0.0 K, avi, 31 Dec 1969)       2015 Video: Glass breakage in slow motion, MIT (0.0 K, 31 Dec 1969)     2009 Video: Glass breakage in slow motion, MIT (same video) (0.0 K, 31 Dec 1969)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 term atom abbreviates Euler 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
Slides: 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)
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.

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
dsolve(de,y(x)); # maple 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.
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 1,2,3,4,5

Presentation 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.Slides: 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.
Video: 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)

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