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2250-4 12:55pm Lecture Record Week 8 S2010

Last Modified: February 28, 2010, 13:33 MST.    Today: September 20, 2018, 08:22 MDT.

Week 8, Mar 1 to 5: Sections 5.1, 5.2, 5.3, 5.4

2 Mar:Basis and dimension.

MAPLE LAB 2. [laptop projection]
   Hand Solution to L2.2.
   Graphic in L2.3.
   Interpretation of graphics in L2.4.
PROBLEM 4.7-26.
    How to solve y''+10y'=0 for general solution y=c1 + c2 exp(-10x)
    Outline of the general theory used to solve linear differential equations.
       Order of a DE and the dimension of the solution space.
       Euler's theorem.
       Finding solution atoms for a basis.
   Definition of basis and span.
   Examples: Find a basis from a general solution formula.
   Bases and the pivot theorem.
   Equivalence of bases.
   A test for equivalent bases.
   THEOREM. Two bases for a vector space V must have the same number of vectors.
     Last Frame Algorithm: Basis for a linear system Ax=0.
     Last frame algorithm and the vector general solution.
     Basis of solutions to a homogeneous system of linear algebraic equations.
     Bases and partial derivatives of the general solution on the invented symbols t1, t2, ...
     DE Example: y = c1 e^x + c2 e^{-x} is the general solution. What's the basis?
    Web References:
    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 (132.6 K, pdf, 03 Feb 2010)
    Manuscript: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)

2 Mar: Introduction to higher order linear DE. Sections 5.1,5.2.

Summary for Higher Order Differential Equations
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.
    Picard's Theorem.
      Solution space structure.
      Dimension of the solution set.
     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 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.
     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.

2 Mar: Constant coefficient equations with complex roots

   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.
  Spring-mass system,
  RLC circuit equation.
  harmonic oscillation,

3-4 Mar: Murphy

Exam 2 review.
Problem sessions on ch4 and ch5 problems. Web 104 Wednesday and JTB 140 Thursday this week
  How to construct solutions to 4.7-10,22,26.
  Questions answered on 4.3, 4.4, 4.5, 4.6, 4.7 problems.
  Survey of 4.3-4.4 problems.
   Illustration: How to do abstract independence arguments using vector
                 packages, without looking inside the packages.
   Applications of the sample test and Wronskian test.
   How to use the pivot theorem to identify independent vectors from a list.

4 Mar: 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.
Euler's theorem.
Quadratic equations again.
Constant-coefficient second order homogeneous differential equations.
Characteristic equation and its factors determine the atoms.

Sample equations:
   y''=0, y''+2y'+y=0, y''-4y'+4y=0,
   y''+y=0, y''+3y'+2y=0,
   mx''+cx'+kx=0, LQ''+RQ'+Q/C=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.

4 Mar: 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 equation,
  LRC-circuit equation,
  Spring-mass DE and RLC-circuit DE derivations.
  Electrical-mechanical analogy.
  forced systems.
  harmonic oscillations,
  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.
  Pure resonance.
  Cafe door.
  Pet door.
  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 (0.0 K, pdf, 31 Dec 1969)
    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)