TopicsSections 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, 24 Dec 2012)

Edwards-Penney 2.1, 2.2, 2.3 (15.5 K, txt, 26 Dec 2012)

Lecture on Section 1.5We 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 1.4-6y' = 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 emptyHow to test separable and non-separable equationsTheorem. 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 methodReferences for linear DE:: Linear integrating factor method (129.8 K, pdf, 03 Mar 2012)Slides: Applications of linear DE, brine tanks, home heating and cooling (374.2 K, pdf, 28 Jul 2009)Manuscript: 1st Order Linear DE part I. Integrating Factor Method, Superposition (152.7 K, pdf, 07 Aug 2009)Manuscript: 1st Order Linear DE part II. Variation of Parameters, Undetermined Coefficients (134.1 K, pdf, 07 Aug 2009)Manuscript: Linear integrating factor method, exercises 1.5-3,5,11,33. Brine mixing (375.0 K, pdf, 29 Jan 2006)Transparencies: How to do a maple answer check for y'=y+2x (0.3 K, txt, 09 Dec 2012)TextLinear 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 TheorySuperposition 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.: 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.TEXTExamples and ApplicationsGrowth-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 1.5-34The 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 concentrationReferences for linear applications: Applications of linear DE (374.2 K, pdf, 28 Jul 2009)Manuscript: Brink tanks (95.3 K, pdf, 04 Mar 2012)Slides: Home heating (109.8 K, pdf, 04 Mar 2012)Slides

Introduction to Ch 2 topics2.1, 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: Maple lab 3 Maple lab 4Drill Section 1.5Drill on 1.5 Problems, as time allowsThere 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.: Linear integrating factor method (129.8 K, pdf, 03 Mar 2012)Slides: Linear DE part I. Integrating Factor Method (152.7 K, pdf, 07 Aug 2009)ManuscriptGeneral Verhulst DE, Section 2.1Solving 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 constantReferences on partial fractions: Partial Fraction Theory (160.7 K, pdf, 03 Mar 2012)Slides: Heaviside coverup partial fraction method (152.1 K, pdf, 07 Aug 2009)Manuscript: Heaviside's method and Laplace theory (186.8 K, pdf, 20 Oct 2009)ManuscriptSeparation of variables and partial fractionsExercise 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)

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.: Autonomous DE (102.3 K, pdf, 03 Mar 2012)Slides: 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)Manuscript: 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)Text: 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)Transparencies: 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)Text: Problem notes S2013 (5.1 K, html, 06 Dec 2012)html

Maple lab 1: quadratics, partial derivatives. Present problem 1 from the midterm 1 sample [S2012 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 S2012. See also F2010, Exam 1.: 2250 midterm exam samples S2013 (19.6 K, html, 30 Apr 2013)HTML

Drill and ReviewPhase 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 modelsIsaac 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 takelonger to rise or longer to fall?: Bolt shot Example 2.3-3 (1.0 K, txt, 17 Dec 2012)Text: Newton kinematics with air resistance. Projectiles. (137.1 K, pdf, 03 Mar 2012)SlidesA rocket from the earth to the moon: Jules Verne Problem (127.0 K, pdf, 10 Dec 2012)SlidesReading 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 examis the S2012 exam. Found at the course web site:: 2250 midterm exam samples S2013 (19.6 K, html, 30 Apr 2013)HTML