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2250-4 12:25pm Lecture Record Week 2 F2010

Last Modified: September 06, 2010, 09:09 MDT.    Today: October 21, 2017, 20:56 MDT.

Week 2, Aug 30 to Sep 3: Sections 1.4, 1.5, 2.1, 2.2.

Aug 30 and 1 Sep: Michal Kordy

Maple lab 1: quadratics, partial derivatives.
Present problem 1 from the midterm 1 sample [S2010 midterm 1 key].
Exam 1 date is in the syllabus and also the online due dates page.
Questions on textbook sections 1.3, 1.4.
Review and drill Ch1.
Sample Exam: Exam 1 key from S2010. See also F2009, Exam 1.

HTML: 2250 midterm exam samples S2010 (0.0 K, html, 31 Dec 1969)

31 Aug: Theory of Linear First Order Differential Equations. Section 1.5.

 Theory of separable equations section 1.4.
  Separation test:
     Define F(x)=f(x,y0)/f(x0,y0),
            G(y)=f(x0,y),
     then FG=f if and only if y'=f(x,y) is separable.
  Non-Separable Test
     f_x/f depends on y ==> y'=f(x,y) not separable
     f_y/f depends on x ==> y'=f(x,y) not separable
   Review: Basic theory of y'=F(x)G(y):
     y(x) = H^(-1)( C1 + int(F)),
     H(u)=int(1/G,u0..u).
     Solutions y=constant are called equilibrium solutions.
       Find them using G(c)=0.
     Non-equilibrium solutions arise from y'/G(y)=F(x) and a
       quadrature step.
Implicit and explicit solutions.
  Discussion of answer checks for implicit solutions and also
     explicit solutions.
  Troubles with explicit solutions of y'= 3 sqrt(xy) [1.4-6].
  Separable DE with no equilibrium solutions.
  Separable DE with infinitely many equilibrium solutions.
  The list of answers to a separable DE.
  Influence of an initial condition to extract just one solution
    formula from the list.
  Examples for Midterm 1 problem 2:
    y'=x+y, y'=x+y^2, y'=x^2+y^2
  Example 1: Show that y'=x+y is not separable using TEST I or II
    (partial derivative tests).
  Example 2: Find the factorization f=F(x)G(y) for y'=f(x,y),
             given
       (1) f(x,y)=2xy+4y+3x+6 [ans: F=x+2, G=2y+3].
       (2) f(x,y)=(1-x^2+y^2-x^2y^2)/x^2 [ans: F=(1-x^2)/x^2, G=1+y^2].
Answer Checks and Key Examples.
  Discussion of answer checks
     implicit solution ln|y|=2x+c for y'=2y
     explicit solution y = C exp(2x) for y'=2y
  Troubles with explicit solutions of y'= 3 sqrt(xy) [1.4-6].
  Key Examples
    Separable DE with no equilibrium solutions.
    Separable DE with finitely many equilibrium solutions.
    Separable DE with infinitely many equilibrium solutions.
  The list of answers to a separable DE.
  Influence of an initial condition to extract just one solution
    formula from the list.
  Examples for Midterm 1 problem 2:
    y'=x+y, y'=x+y^2, y'=x^2+y^2
Lecture on Section 1.5
  Theory of linear DE y'=-p(x)y+q(x).
  Integrating factor W=e^Q(x), Q(x) = int( p(x),x)
  (Wy)'/W, the fraction that replaces two-termed expression y'+py.
Classification of y'=f(x,y)
    quadrature [Q], separable [S], linear [L].
    Venn diagram of classes Q, S, L.
    Examples of various types.
    Test for quadrature (f_y=0)
    Test for linear (f_y indep of y)
    Test for not separable (f_y/f depends on x ==> not sep)
    Finding F and G in a separable equation y'=F(x)G(y)
  
Linear integrating factor method 1.5
  Application to y'+2y=1 and y'+y=e^x.
  Examples:
    Testing linear DE y'=f(x,y) by f_y independent of y.
    Classifying linear equations and non-linear equations.
  Picard's theorem implies a linear DE has a unique solution.
  Main theorem on linear DE and explicit general solution.
    References for linear DE:
    Slides: Linear integrating factor method (99.4 K, pdf, 20 Jan 2008)
    Transparencies: Linear DE method, 1.5-3,5,11,33. Brine mixing (375.0 K, pdf, 29 Jan 2006)
    Manuscript: Applications of linear DE (374.2 K, pdf, 28 Jul 2009)
    Manuscript: Linear DE part I. Integrating Factor Method (152.7 K, pdf, 07 Aug 2009)
    Manuscript: Linear DE part II. Variation of Parameters, Undetermined Coefficients (134.1 K, pdf, 07 Aug 2009)
    Text: How to do a maple answer check for y'=y+2x (0.2 K, txt, 27 Jan 2005)
    Slides: Variation of Parameters. Integrating factor method (24.6 K, pdf, 23 Jan 2007)

31 Aug: Linear Applications. Section 1.5

Review and Drill Section 1.4
  Variables separable method.
  Discuss remaining exercises 1.4-6,12,18.
    Problem Notes 1.4 at the web site.
  Equilibrium solutions and how to find them.
Review and Drill
  Method of Quadrature
   Variables Separable method
     Equilibrium solutions from G(y)=0 and
     Non-equilibrium solutions from G(y) nonzero.
Detailed derivations for 1.4-6
    y' = 3 sqrt(-x) sqrt(-y)  on quadrant 3, x<0, y<0
    y' = 3 sqrt(x) sqrt(y)  on quadrant 1, x>0, y>0
    Equilibrium solution
      Found by sybstitution of y=c into the DE y'=3 sqrt(xy)
      Ans: y=0 is an equilibrium solution
    Non-equilibrium solution
      Found from y'=F(x)G(y) by division by G(y),
        followed by the method of quadrature.
      Applied to quadrant 1
         y = ( x^(3/2)+c)^2
      Applied to quadrant 3
         y = - ((-x)^(3/2)+c)^2
    List of 3 solutions cannot be reduced in number
    Graphic showing threaded solutions: quadrants 2,4 empty

How test separable and non-separable equations
   Theorem. If f_y/f depends on x, then y'=f(x,y) is not separable
   Theorem. If f_x/f depends on y, then y'=f(x,y) is not separable
   Theorem. If y'=f(x,y) is separable, then f(x,y)=F(x)G(y) is
            the separation, where F and G are defined by the formulas
               F(x) = f(x,y0)/f(x0,y0)
               G(y) = f(x0,y).
            The invented point (x0,y0) may be chosen conveniently,
            subject to f(x0,y0) nonzero.
Partial fractions
   How to solve y'=(1-y)y
   Def: a partial fraction = constant / polynomial with one root
   Theorem. A polynomial quotient p(x)/q(x) is a sum of partial
            fractions, provided degree(p) < degree(q).
            The possible partial fractions have denominator
            dividing the denominator of q(x).
   How to solve for partial fraction constants A,B,C,...
      Clear the fractions
      Substitute invented values for x to get a system of equations
        for A,B,C,..., then solve the system.
      Methods:
         Sampling method [described above]
         Method of atoms [multiply out, match powers]
         Heaviside's coverup method
 
General Verhulst DE
    Solving y'=(a-by)y by a substitution
       Let u=y/(a-by).
       Then substitution into the DE gives u'=au
       Solve u'=au to get u=u0 exp(ax).
       Back-substitute u(x) into u=y/(a-by), then solve for y.
   Solving y'=(a-by)y by partial fractions
       Divide the DE by (a-by)y
       Apply the method of quadrature.
       Find the constants in the partial fractions on the left.
       Integrate to get the answer
                       a y0
       y(x) = --------------------------
              b y0 + (a - b y0) exp(-ax)
       where y0=y(0)=initial population size.
   
Linear Differential Equation y'+p(x)y=q(x)
   Section 1.5
     Definition: Linear DE
     Test: y'=f(x,y) is linear if and only if the partial
           derivative f_y is independent of y.
     Algorithm
       Test the DE for linear
       Identify p(x), q(x) in the standard form y'+py=q.
       Determine an integrating factor W(x)=exp(int(p(x)dx))
       Replace y'+py in the standard form y'+py=q by the quotient
          (Wy)' / W
       and then clear fractions to get the quadrature equation
           (Wy)' = qW
       Solve by the method of quadrature.
       Divide by W to find an explicit solution y(x).
   Three linear examples: y'+(1/x)y=1, y'+y=e^x, y'+2y=1.
   classification: separable, quadrature, linear.
   Two Methods for solving first order equations:
     Linear integrating factor method,
     Superposition + equilibrium solution for
         constant-coefficient linear,

    References for linear applications
    Manuscript: Applications of linear DE (374.2 K, pdf, 28 Jul 2009)
    Slides: Brink tanks (62.9 K, pdf, 30 Nov 2009)
    Slides: Home heating (73.8 K, pdf, 30 Nov 2009)

02 Sep: Autonomous systems and applications section 2.1

Superposition Theory
  Superposition for y'+p(x)y=0.
  Superposition for y'+p(x)y=q(x)
  A faster way to solve y'+2y=1
Problem 1.5-34
    The expected model is
      x'=1/4-x/16,
      x(0)=20,
    using units of millions of cubic feet.
  The answer is x(t)=4+16 exp(-t/16).
  Model Derivation
    Law:  x'=input rate - output rate.
    Definition:  concentration == amt/volume.
    Use of percentages
       0.25% concentration means 0.25/100 concentration
Drill Section 1.5
   Three linear examples: y'+(1/x)y=1, y'+y=e^x, y'+2y=1.
   classification: separable, quadrature, linear.
   Methods for solving first order equations:
     Linear integrating factor method,
     Superposition + equilibrium solution for
         constant-coefficient linear DE
   Drill: worksheet distributed in class, for the example
   y' + 2y = 6. Solved in class y'+3y=6, y'+y=e^x, and several
   homogeneous equations like y'+3y=0, y'+2y=0. Solved for
   equilibrium solutions in more complicated examples like
   2y' + Pi y = e^2.

02 Sep: Autonomous Differential Equations and Phase Diagrams. Sections 2.1, 2.2

Problem 1.5-34
    The expected model is
      x'=1/4-x/16,
      x(0)=20,
    using units of millions of cubic feet.
  The answer is x(t)=4+16 exp(-t/16).
  Model Derivation
    Law:  x'=input rate - output rate.
    Definition:  concentration == amt/volume.
    Use of percentages
       0.25% concentration means 0.25/100 concentration

html: Problem notes F2010 (4.6 K, html, 26 Nov 2010)
Examples and Applications
  Growth-Decay model y'=ky and its algebraic model y=y(0)exp(kx).
   Pharmokinetics of drug transport [ibuprofen]
   Pollution models.
     Three lake pollution model [Erie, Huron, Ontario].
   Brine tanks.
     One-tank model.
     Two-tank and three-tank models.
     Recycled brine tanks and limits of chapter 1 methods.
   Linear cascades and how to solve them.
     Method 1: Linear integrating factor method.
     Method 2: Superposition and equilibrium solutions for
       constant-coefficient y'+py=q. Uses the shortcut for
       homogeneous DE y'+py=0.
   Separation of variables
      The equation y'=7y(y-13), y(0)=17
      F(x) = 7, G(y) = y(y-13)
      Separated form y'/G(y) = F(x)
      Answer check using the Verhulst solution
          P(t) = aP_0/(bP_0 + (a-b P_0)exp(-at))
      Separation of variables details.
      Review of partial fractions.
      Partial fraction details for 1/((u(u-13)) = A/u + B/(u-13)
  Partial fractions.
   DEFINITION: partial fraction=constant/polynomial with exactly one root
   THEOREM: P(x)/q(x) = a sum of partial fractions
   Finding the coefficients.
     Method of sampling
       clear fractions, substitute samples, solve for A,B, ...
     Method of atoms
       clear fractions, multiply out and match powers, solve for A,B,...
     Heaviside's cover-up method
       partially clear fraction, substitute root, find one constant
  Separation of variable solutions with partial fractions.
  Exercise solutions to the problems due in 2.1.
    References for 2.1, 2.2, 2.3. Includes the rabbit problem, partial fraction examples, phase diagram illustrations.
    Slides: Autonomous DE (69.9 K, pdf, 03 Sep 2009)
    Manuscript: Verhulst logistic equation (115.5 K, pdf, 02 Oct 2009)
    Manuscript: Phase Line and Bifurcation Diagrams. Includes Stability, Funnel, Spout, and bifurcation (227.4 K, pdf, 07 Sep 2009)
    Transparencies: ch2 sections 1,2,3: 2.1-6,16,38, 2.2-4,10, 2.3-9,27+Escape velocity (357.6 K, pdf, 29 Jan 2006)
    Text: ch2 DEplot maple example 1 for exercises 2.2, 2.3 (0.7 K, txt, 07 Sep 2009)
    Text: ch2 DEplot maple example 2 for exercises 2.2, 2.3 (0.7 K, txt, 07 Sep 2009)
    References for linear applications
    Manuscript: Applications of linear DE (374.2 K, pdf, 28 Jul 2009)
    Slides: Brink tanks (62.9 K, pdf, 30 Nov 2009)
    Slides: Home heating (73.8 K, pdf, 30 Nov 2009)
    Manuscript: Systems theory and examples (785.8 K, pdf, 16 Nov 2008)
Midterm 1 sample exam is the S2010 exam, found at the course web site.
HTML: 2250 midterm exam samples F2010 (17.7 K, html, 18 Dec 2010)