2250 Lecture Record Week 8 F2009
Last Modified: October 25, 2009, 20:03 MDT.    Today: December 10, 2017, 23:07 MST.
Week 8, Oct 19 to 23: Sections 5.1,5.2,5.4,5.4,10.1
19 Oct: Introduction to higher order linear DE. Sections 5.1,5.2.
References from before Fall break
Summary for Higher Order Differential Equations
Slides: Orthogonality, CSB-inequality, Pythagorean identity (87.2 K, pdf, 10 Mar 2008)
Slides: The pivot theorem and applications (131.9 K, pdf, 02 Oct 2009)
Slides: Matrix add, scalar multiply and matrix multiply (122.5 K, pdf, 02 Oct 2009)
Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (153.7 K, pdf, 16 Oct 2009)
Slides: More on digital photos, checkerboard analogy (109.5 K, pdf, 02 Oct 2009)
Slides: Vector space, subspace, independence (132.5 K, pdf, 03 Oct 2009)
Manuscript: Vector space, Independence, Basis, Dimension, Rank (206.4 K, pdf, 27 Feb 2007)
Slides: Atoms, Euler's theorem, 7 examples (96.6 K, pdf, 20 Oct 2009)
Slides: Base atom, atom, basis for linear DE (85.4 K, pdf, 20 Oct 2009)
EXAMPLE. The equation y'' 10y'=0.
How to solve y'' + 10y' = 0 with chapter 1 methods. Midterm 1 problem 1(d).
Idea: Let v=x'(t) to get a first order DE in v and a quadrature equation x'(t)=v(t).
Solve the first order DE by the linear integrating factor method. Then insert
the answer into x'(t)=v(t) and continue to solve for x(t) by quadrature.
Vector space of functions: solution space of a differential equation.
A basis for the solution space of y'' + 10y'=0.
Base atoms are 1, exp(a x), cos(b x), sin(b x), exp(ax)cos(bx), exp(ax)sin(bx).
Define: atom=x^n(base atom).
THEOREM. Atoms are independent.
THEOREM. Solutions of constant-coefficient homogeneous differential
equations are linear combinations of atoms.
PICARD 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) are a basis for
the solution space of the differential equation.
Then the general solution is
y = c1 (1) + c2 (exp(-10x)).
Survey of topics for this week.
Linear DE Slides.
Slides: Picard-Lindelof, linear nth order DE, superposition (121.7 K, pdf, 18 Oct 2009)
Slides: How to solve linear DE or any order (104.1 K, pdf, 18 Oct 2009)
Slides: Atoms, Euler's theorem, 7 examples (96.6 K, pdf, 20 Oct 2009)
Theory of Higher Order Constant Equations:
Homogeneous and non-homogeneous structure.
Solution space structure.
Dimension of the solution set.
Definition of atom.
Independence of atoms.
Non-real roots [complex roots].
How to deal with conjugate pairs of factors (r-a-ib), (r-a+ib).
The 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.
Harmonic oscillator example y'' + y=0.
Dimension of the solution space.
Structure of solutions.
Non-homogeneous equation. Forcing term.
Nth order equations.
Solution space theorem for linear differential equations.
Independence and Wronskians. Independence of atoms.
Main theorem on constant-coefficient equations [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 homogeneous differential equation.
Picard's Theorem for higher order DE and systems.
20 Oct: Constant coefficient equations with complex roots
Visualization of matrix add and matrix scalar multiply.
Chapter 4 exercises.
Lecture: Constant coefficient equations with complex roots.
How to solve for atoms when the characteristic equation has multiple roots or complex roots.
Applying Euler's theorems to solve a DE.
Examples of order 2,3,4. Exercises 5.1, 5.2, 5.3.
RLC circuit equation.
21 Oct: Second order and higher order differential Equations
Lecture: Second order and higher order differential Equations.
Picard theorem for second order equations, superposition, solution
space structure, dimension of the solution set.
Quadratic equations again.
Constant-coefficient second order homogeneous differential equations.
Characteristic equation and its factors determine the atoms.
y''=0, y''+2y'+y=0, y''-4y'+4y=0,
Solved 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 complex root.
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.
Review and Drill.
The RLC circuit equation and its physical parameters.
Spring-mass equation mx''+cx'+kx=0 and its physical parameters.
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.
23 Oct: Damped and Undamped Motion. Intro Laplace Theory. Sections 5.4, 10.1
Lecture: Applications. Damped and undamped motion.
Theory of equations and 5.3-32.
Problems discussed in class: all of 5.2, 5.3 and 5.4-20,34
Spring-mass DE and RLC-circuit DE derivations.
phase-amplitude conversions from the trig course.
Damped and undamped equations. Phase-amplitude form.
Partly 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 ==> omega. Because the two roots a+ib, a-ib
determine the quadratic equation, then c and k are known in terms
of symbols a,b.
Over-damped, Critically-damped and Under-damped behavior,
References: Sections 5.4, 5.6. Forced oscillations.
Slides: Unforced vibrations 2008 (620.4 K, pdf, 11 Oct 2009)
Slides: Forced undamped vibrations (174.7 K, pdf, 11 Oct 2009)
Slides: Forced damped vibrations (235.0 K, pdf, 11 Oct 2009)
Slides: Forced vibrations and resonance (185.3 K, pdf, 11 Oct 2009)
Slides: Resonance and undetermined coefficients, cafe door, pet door, phase-amplitude (143.3 K, pdf, 07 Nov 2009)
Slides: Electrical circuits (87.1 K, pdf, 11 Oct 2009)
Lecture: Introduction to Laplace theory.
Newton and Laplace: portraits of the Two Greats [slides].
Method of quadrature.
Comparison of Newton calculus and Laplace calculus.
Direct Laplace transform == Laplace integral.
Laplace theory references
Slides: Laplace and Newton calculus. Photos of Newton and Laplace. (145.3 K, pdf, 01 Nov 2009)
Manuscript: Heaviside's method with Laplace Examples 2008 (186.8 K, pdf, 20 Oct 2009)
Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1968.3 K, pdf, 13 Nov 2003)
Manuscript: Laplace theory 2008 (350.5 K, pdf, 06 Mar 2009)
Slides: Laplace examples (101.2 K, pdf, 07 Nov 2009)
Slides: Intro to Laplace theory. Assumes only calculus (109.5 K, pdf, 01 Nov 2009)
Slides: Laplace rules (112.2 K, pdf, 01 Nov 2009)
Slides: Laplace table proofs (130.3 K, pdf, 01 Nov 2009)
MAPLE: Maple Lab 7. Laplace applications (84.3 K, pdf, 19 Jul 2009)
Slides: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)
Slides: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)
Slides: Piecewise functions and Laplace theory (64.7 K, pdf, 01 Nov 2009)