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2250 7:30am Lectures Week 2 S2014

Last Modified: January 17, 2014, 06:13 MST.    Today: September 24, 2018, 02:15 MDT.
  Sections 1.5, 2.1, 2.2, 2.3
  The textbook topics, definitions, examples and theorems
Edwards-Penney 1.2, 1.3, 1.4, 1.5 (14.7 K, txt, 19 Dec 2013)
Edwards-Penney 2.1, 2.2, 2.3 (15.5 K, txt, 17 Jan 2014)

Week 2, Jan 13 to 17

Sections 1.5, 2.1, 2.2, 2.3

Monday: Solving Linear DE. Section 1.5.

Lecture on Section 1.5
  We will study linear DE y'=-p(x)y+q(x).
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)
Section 1.4. Separable DE. Review and Drill, as time allows.
  Variables separable method.
    Finding F and G in a separable equation y'=F(x)G(y)
      Equilibrium solutions from G(y)=0 and
      Non-equilibrium solutions from G(y) nonzero.
      Method of Quadrature: When to use it.
  Discuss remaining exercises 1.4-6,12,18.
    Problem Notes 1.4 at the web site.
  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 substitution 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 shows threaded solutions: quadrants 2,4 empty
How to 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.
Section 1.5. Linear integrating factor method
References for linear DE: 
Slides: Linear integrating factor method (129.8 K, pdf, 03 Mar 2012)
Manuscript: Applications of linear DE, brine tanks, home heating and cooling (484.2 K, pdf, 16 Jan 2014)
Manuscript: 1st Order Linear DE part I. Integrating Factor Method, Superposition (303.6 K, pdf, 17 Jan 2014)
Manuscript: 1st Order Linear DE part II. Variation of Parameters, Undetermined Coefficients (238.7 K, pdf, 17 Jan 2014)
Manuscript: Kinetics, Newton's Models (343.8 K, pdf, 17 Jan 2014)
Transparencies: Linear integrating factor method, exercises 1.5-3,5,11,33. Brine mixing (375.0 K, pdf, 29 Jan 2006)
Text: How to do a maple answer check for y'=y+2x (0.3 K, txt, 07 Jan 2014) Linear Differential Equation y'+p(x)y=q(x) Section 1.5 Definition: Linear DE y'+p(x)y=q(x) Test: y'=f(x,y) is linear if and only if the partial derivative f_y is independent of y. 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. We don't check for equilibrium solutions or exceptions. THEOREM. A linear DE has an explicit general solution. Def. Integrating factor W=exp(Q(x)), where Q(x) = int(p(x),x) THEOREM. The integrating factor fraction (Wy)'/W replaces the two-termed expression y'+py. Application Examples: y'+2y=1 and y'+y=e^x. ALGORITHM. How to solve a linear differential equation 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),x)) 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. Superposition Theory Superposition for y'+p(x)y=0. Superposition for y'+p(x)y=q(x) Methods for solving first order linear equations: Failsafe: Linear integrating factor method, Shortcut: Superposition + equilibrium solution for constant-coefficient linear DE Fastest way to solve y'+2y=1 is the Shortcut. Drill: Sample worksheet distributed in class, for the example y' + 2y = 6 and more.
TEXT: Examples 1st order linear DE, integrating factor methods (3.2 K, txt, 17 Dec 2012) Solved in class y'+3y=6, y'+y=e^x, and homogeneous equations like y'+3y=0, y'+2y=0. Solve for equilibrium solutions in strange examples like 2y' + Pi y = e^2. Examples and Applications Growth-Decay model y'=ky and its algebraic model y=y(0)exp(kx). Pharmacokinetics of drug transport [ibuprofen] Pollution models. Three lake pollution model [Erie, Huron, Ontario]. Brine tanks. One-tank model. Two-tank and three-tank models. Chemical engineering example, 3 tanks. 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. Problem 1.5-34 The expected model is x'=1/4-x/16, x(0)=20, using units of days and 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. Example: 2kg salt in 15 liters brine, conc=(2/15)kg/liter Use of percentages 0.25% concentration means 0.25/100 concentration References for linear applications
Manuscript: Applications of linear DE (484.2 K, pdf, 16 Jan 2014)
Slides: Brink tanks (95.3 K, pdf, 04 Mar 2012)
Slides: Home heating (99.3 K, pdf, 10 Apr 2014)

Tuesday: Linear DE. Autonomous systems Section 2.1

Introduction to Ch 2 topics
  2.1, 2.2: Autonomous DE y'=f(y)
    Solution of the Verhulst DE y'=(a-by)y
  2.3: Newton models, Jules Verne problem
  2.4,2.5.26: Numerical solutions of DE.
     No exercises, but a numerical project based on EPH exercises.
Drill on 1.5 Problems, as time allows
  There are two special methods for solving y'+py=q
     If p,q are constant then use the SUPERPOSITION METHOD
         y = y_p + y_h
         y_p = an equilibrium solution (set y'=0, solve for y)
         y_h = constant divided by the integrating factor
     If one of p or q depends on x, then use the STANDARD METHOD
         Replace the LHS, which is y'+p(x)y, by the integrating
         factor quotient (Wy)'/W, where W=exp(int p(x)dx)) is the
         integrating factor.
         Cross-multiply by W to clear fractions. Then apply the
         method of quadrature.  
Slides: Linear integrating factor method (129.8 K, pdf, 03 Mar 2012)
Manuscript: Linear DE part I. Integrating Factor Method (303.6 K, pdf, 17 Jan 2014) General Verhulst DE, Section 2.1 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. Review of 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 References on partial fractions
Slides: Partial Fraction Theory (160.7 K, pdf, 03 Mar 2012)
Manuscript: Heaviside coverup partial fraction method (290.2 K, pdf, 07 Jan 2014)
Manuscript: Heaviside's method and Laplace theory (352.3 K, pdf, 07 Jan 2014) Separation of variables and partial fractions Exercise solution problem 2.1-8 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. Partial fraction details for 1/((u(u-13)) = A/u + B/(u-13)

Wednesday: Autonomous Differential Equations and Phase Diagrams. Section 2.2

Lecture on 2.2:
  Theory of autonomous DE y'=f(y)
     Picard's theorem and non-crossing of solutions.
     Direction fields and translation of solutions
  Constructing Euler's threaded solution diagrams
     No direction field is needed to draw solution curves
       We throw out the threaded solution rule used in chapter 1,
       replace it by two rules from calculus and a theorem:
          1. If y'(x)>0, then y(x) increases.
          2. If y'(x)<0, then y(x) decreases.
          THEOREM. For y'=f(y), a threaded solution starting with
              y'(0)>0 must satisfy y'(x)>0 for x>0. A similar result
              holds for y'(0)<0.
     Definition: phase line diagram, phase diagram,
       Calculus tools: f'(x) pos/neg ==> increasing/decreasing
       DE tool: solutions don't cross
       Maple tools for production work.
  Stability theory of autonomous DE y'=f(y)
    Stability of equilibrium solutions.
    Stable and unstable classification of equilibrium solutions.
    funnel, spout, node,
  How to construct Phase line diagrams
  How to make a phase diagram graphic
    Inventing a graph window
    Invention of the grid points
    Using the phase line diagram to make the graphic
        calculus tools
        DE tools
 References for 2.1, 2.2, 2.3. Includes the rabbit problem, partial
 fraction examples, phase diagram illustrations.
Slides: Autonomous DE (101.8 K, pdf, 15 Jan 2014)
Manuscript: Verhulst logistic equation (115.5 K, pdf, 02 Oct 2009)
Manuscript: fish farming (384.5 K, pdf, 16 Jan 2014)
Manuscript: Phase Line and Bifurcation Diagrams. Includes Stability, Funnel, Spout, and bifurcation (765.5 K, pdf, 15 Jan 2014)
Text: Problem notes 2.1 (2.6 K, txt, 06 Dec 2012)
Text: Problem notes 2.2 (3.4 K, txt, 06 Dec 2012)
Text: Problem notes 2.3 (2.8 K, txt, 05 Dec 2012)
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)
html: Problem notes S2014 (4.9 K, html, 10 Dec 2013)

Thursday: Rebecca

Second Lab project.

Friday: Newton Kinematic Models. Projectiles. Jules Verne. Section 2.3.

Drill and Review
  Phase diagram for y'=y^2(y^2-4)
     Phase line diagram
     Threaded curves
     Labels: stable, unstable, funnel, spout, node

  Phase line diagrams.
  Phase diagram.
Newton's force and friction models
  Isaac Newton ascent and descent kinematic models.
    Free fall with no air resistance F=0.
    Linear air resistance models F=kx'.
    Non-linear air resistance models F=k|x'|^2.
The tennis ball problem.
  Does it take longer to rise or longer to fall?
Text: Bolt shot Example 2.3-3 (1.0 K, txt, 17 Dec 2012)
Slides: Newton kinematics with air resistance. Projectiles. (137.1 K, pdf, 03 Mar 2012) A rocket from the earth to the moon
Slides: Jules Verne Problem (127.0 K, pdf, 10 Dec 2012) Reading assignment: Proofs of 2.3 theorems in the textbook and derivation of details for the rise and fall equations with air resistance.
Problem notes for 2.3 (2.8 K, txt, 05 Dec 2012) Midterm 1 sample exam is is to be written. Found at the course web site:
HTML: 2250 midterm exam samples S2014 (22.4 K, html, 30 Apr 2014)