2250-1 7:30am Lecture Record Week 2 S2010

Last Modified: January 24, 2010, 09:43 MST.    Today: October 23, 2017, 21:44 MDT.

Week 2, Jan 18 to 22: Sections 1.4, 1.5, 2.1, 2.2.

19 Jan: Theory of Linear First Order Differential Equations. Section 1.5.

Topics delayed from Week 1.
  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
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.
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)

20 Jan: Linear Applications. Section 1.5

Collect first package of dailies in class: 1.2, 1.3.
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 to deal with 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)

20-21 Jan: Murphy

Present problems 2, 3 of the midterm 1 sample [f2009 midterm 1 key].
Exam 1 date is 17 Feb 1-3pm in WEB 104 or 18 Feb 6:50am in JTB 140. Other exam times were pre-set by agreement at the start of the semester, on an individual basis. The plan was created to provide extra time to write the exam, which is designed for 50 min.
Sample Exam: Exam 1 key from F2009. See also S2009, Exam 1.
HTML: 2250 midterm exam samples S2010 (15.6 K, html, 16 May 2010)
Questions on textbook sections 1.3, 1.4.
Review and drill Ch1.

22 Jan: 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.
    References for 2.1, 2.2:
    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)
Midterm 1 sample exam is the F2009 exam, found at the course web site.
HTML: 2250 midterm exam samples S2010 (15.6 K, html, 16 May 2010)
To date, Murphy has covered problems 1,2,3 in the exam review sessions on Wed-Thu.