## 2250 7:30am Lectures Week 9 S2013

Last Modified: March 17, 2013, 20:19 MDT.    Today: July 23, 2018, 03:56 MDT.
``` Edwards-Penney, sections 5.4, 10.1, 10.2, 10.3
The textbook topics, definitions and theoremsEdwards-Penney 5.1, 5.2, 5.3, 5.4 (15.6 K, txt, 23 Dec 2012)Edwards-Penney 10.1, 10.2, 10.3, 10.4, 10.5 (20.5 K, txt, 03 Mar 2013)```

#### Monday: Applications x'' + px' + qx=0. Section 5.4.

```PROBLEM SESSION
Chapter 4 exercises.
4.7 Independence. Classify as indep/dep. Cite these results.
THEOREM.
Any set of distinct Euler solution atoms is independent.
THEOREM.
Wronskian determinant nonzero => functions in Wronskian row 1
are independent.
THEOREM.
Subsets of independent sets are independent.
THEOREM.
S1 is a finite list of k vectors.
S2 is a finite list of k vectors (the same number).
span(S1) = span(S2)
If S2 is independent, then S1 is also independent.

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
[This is S1. Verify S2 = 1,x,x^2 works in the theorem.]
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
[This is S1. What is S2?]
```
```Second order and higher order differential Equations.
Application. LRC circuit equation.
LQ'' + RQ' + (1/C)Q = 0
LI'' + RI' + I/C = 0.
The RLC circuit diagram and the physical parameters.
Forcing terms in electrical systems. Battery. Generator.
LRC-circuit DE derivation.
Voltage drop laws of Faraday, Ohm, Coulomb.
Kirchoff's laws.
Electrical-mechanical analogy. Slides: Electrical circuits (124.8 K, pdf, 04 Mar 2012)
Detailed look at the second order case
REF. Theorems 1,2,3 in section 5.3 of Edwards-Penney.
Constant-coefficient second order homogeneous differential equations.
Characteristic equation and its factors determine the atoms.
The three possible cases
1. Two real roots
2. Two equal real roots
3. A complex conjugate pair of roots
The three possible kinds of solutions
1. y=c1 exp(r1 x) + c2 exp(r2 x)
If one root is zero, then the exponential equals 1.
2. y=c1 exp(r1 x) + c2 x exp(r1 x)
3. y= c1 exp(ax) cos(bx) + c2 exp(ax) sin(bx)
If in root a+ib, the real part a=0, then this is a pure
harmonic oscillation [because exp(ax)=1]
y= c1 cos(bx) + c2 sin(bx)
Sample equations:
y''=0, y''+2y'+y=0, y''-4y'+4y=0, y'' + 3y' + 2y=0,
x'' + x = 0, x'' + 2x' + 5x = 0, x'' + 8x' + 16x=0,
0.03I'' + 0.005 I = 0, Q'' + 100Q=0,

REVIEW OF SECTIONS 5.1, 5.2, 5.3
PROBLEM TYPES.
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
are known to be solutions.
Example: Construct a linear DE from characteristic equation roots.
Example: Construct a linear DE from its general solution.

COMPLEX ROOTS of the CHARACTERISTIC EQUATION
Solving a DE when the characteristic equation has complex roots.
Equations with both real roots and complex roots.
(D^4 + D^2)y = 0, (D+1)^2(D^2+4)^2 y = 0
An equation with 4 complex roots. How to find the 4 atoms.
(D^2+4)(D^2+16)y=0
SHORTCUT. One pair of complex conjugate roots identifies two Euler
solution atoms. Only one of the two complex roots is required to
construct the two atoms.

Drill Partial Fractions
top=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.
Maple example with 6 constants
top:=x-1; bottom:=(x+1)^2*(x^2+1)^2;
convert(top/bottom,parfrac,x);
```
```Slides on Section 5.4
Damped oscillations
overdamped, critically damped, underdamped,
pseudo-period  [Chapter 5]
phase-amplitude form of the solution [chapter 5]
Cafe door.
Pet door.
Undamped oscillations.
Harmonic oscillator. Slides: phase-amplitude, cafe door, pet door, damping classification (148.4 K, pdf, 03 Mar 2013)
Partly solved 5.4-20
See the FAQ at the web site for answers and details.Text: FAQ for section 5.4 (4.7 K, txt, 05 Dec 2012)
The 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.
The Euler solution atoms are
cos( sqrt(20)t), sin( sqrt(20)t).
UNDAMPED HARMONIC OSCILLATION.

Each system has general solution a linear combination of Euler
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)
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)/2

Partly solved 5.4-34.
See the FAQ at the web site for answers and details.Text: FAQ for section 5.4 (4.7 K, txt, 05 Dec 2012)
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.

References:  Sections 5.4, 5.6. Forced oscillations.
Slides: Unforced vibrations 2008 (667.9 K, pdf, 03 Mar 2012) Slides: phase-amplitude, cafe door, pet door, damping classification (148.4 K, pdf, 03 Mar 2013) Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012) Slides: Forced damped vibrations (280.7 K, pdf, 04 Mar 2012) Slides: Forced vibrations and resonance (240.3 K, pdf, 04 Mar 2012) Slides: Undetermined coefficients, pure resonance, practical resonance (152.8 K, pdf, 03 Mar 2013) Slides: Electrical circuits (124.8 K, pdf, 04 Mar 2012)```

#### Tuesday: Intro to Laplace Theory. Sections 10.1,10.2,10.3.

```More exam 2 review, problems 4,5
Lecture: Basic Laplace theory.
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 and History of Laplace's method
Photos of Newton and Laplace: portraits of the Two Greats.Slides: Laplace and Newton calculus. Photos of Newton and Laplace. (200.2 K, pdf, 04 Mar 2012)
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 abbreviation L(f(t)) for the Laplace integral of f(t).
Lerch's cancelation law and the fundamental theorem of calculus.
Intro to Laplace TheorySlides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 19 Mar 2012)
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

Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 04 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 19 Mar 2012) Manuscript: Laplace theory 2008 (351.3 K, pdf, 09 Apr 2013) Slides: Laplace rules (160.3 K, pdf, 04 Mar 2012) Slides: Laplace table proofs (169.6 K, pdf, 04 Mar 2012) Slides: Laplace examples (149.1 K, pdf, 04 Mar 2012) Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013) MAPLE: Maple Lab 7. Laplace applications (156.0 K, pdf, 04 Dec 2012) Slides: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012) Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008) Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008) Manuscript: Heaviside's method 2008 (186.8 K, pdf, 20 Oct 2009) Manuscript: DE systems, examples, theory (785.8 K, pdf, 16 Nov 2008) Manuscript: Laplace theory 2008 (351.3 K, pdf, 09 Apr 2013) Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 28 Nov 2010) Text: Laplace theory problem notes S2013 (17.2 K, txt, 03 Dec 2012) Text: Final exam study guide (8.2 K, txt, 05 Dec 2012) Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012)```

#### Wednesday: Laplace Theory. Tables, Rules of Laplace's Calculus 10.2, 10.3.

``` A brief Laplace table.
Forward table.
Backward table.
Extensions of the Table.
Laplace rules.
Linearity.
The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
Shift theorem.
Parts theorem.
Finding Laplace integrals using Laplace calculus.
Solving differential equations by Laplace's method.
Basic Theorems of Laplace Theory
Functions of exponential order
Existence theorem for Laplace integrals
Euler solution atoms have a Laplace integral
Lerch's theorem
Linearity.
The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
Shift theorem L(exp(at)f(t)) = L(f(t))|s->(s-a)
Parts theorem L(y')=sL(y)-y(0)
```

#### Thursday:

``` Exam 2 review and Problem session on ch5 problems.
Exam 2 review for problems 1,2,3,4,5.
```

#### Friday: Laplace Theory. Applications of Laplace's method from 10.3, 10.4, 10.5

``` History of the Laplace Transform
REF: Deakin (1981), Development of the Laplace transform 1737 to 1937
EULER
LAPLACE 1784
End of WWII 1945
Fourier Transform
Mellin Transform and Gamma function
Laplace transform: one-sided and 2-sided transform
Applications: DE, PDE, difference equations, functional equations
Diffusion equation for spatial diffusion problems

Solving y' = -1, y(0)=2 [done on Wednesday]

DEF. Gamma function
DEF. Unit step u(t-a)=1 for t>=a, else zero
DEF. Ramp t->(t-a)u(t-a)
Backward table problems: examples
Forward table problems: examples
Computing Laplace integrals L(f(t)) with rules
Solving an equation L(y(t))=expression in s for y(t)