Math 225-Section 01

Introduction to Differential Equations

Spring Semester 2008

• M225-01 Web Page: http://www.math.montana.edu/~vogel/Courses/M225_008/M225.html
• Class Meets: 9:00-9:50 am, MF in 307 Roberts Hall; TuTh in 210 RH
• Instructor:
  Curt Vogel
  2-210 Wilson Hall
  ph. 994-5332
  e-mail: vogel (at) math.montana.edu
  Office Hours: 10:00-10:50 MTuThF
• Textbook
  Elementary Differential Equations, 8th Edition, by W.E. Boyce and R.C. DiPrima
  published by John Wiley and Sons, 2005
• Grading
  1/3 Homework and Quizes
  1/3 In-Class Exams (2)
  1/3 Final Exam
• Computer Software
  Click here to access a Rice University web site that has java versions of dfield and pplane. dfield can be used to plot direction fields and integral curves for first order ODEs (see sections 1.2-1.3). pplane can be used to generate phase portraits for 2-component systems of autonomous ODEs (see chapter 9). Java applets allow you to run the Rice University software over the web on your laptop or PC.
• Thurs, 1/17/08: Section 1.1-1.3. Basic Models & Classification
• Fri, 1/18/08: Section 1.1-1.3. Direction Fields and Solutions
• Quiz 1: Fri, 1/18: Basic Differentiation and Integration. Click here to download solutions in PDF format.
• Mon, 1/21/08: MLK Day---No Class
• Tues, 1/22/08: Elementary Solution Methods. Click here to download handout in PDF format. Recommended HW: Sect. 1.2, p. 15, #1ab,8ab,15ab,16
• Thurs, 1/24/08: Qualitative behavior & mathematical software. Recommended HW: Sect 1.1, p. 7, #1,3,7,9,11,12,13
• Fri, 1/25/08: Method of variation of parameters. This is an alternative to the method of integrating factors, presented in sect 2.1. Click here to download handout in PDF format. Recommended HW: Sect 2.1, p. 39, #1,5,7,19,21.
• Quiz 2: Fri, 1/25: Elementary solution methods (direct integration and separation of variables) Click here to download solutions in PDF format.
• Mon, 1/28/08: Modeling with 1st Order Eqns. Recommended HW: Sect 2.3, p. 59, #1,3,6,9,12
• Tues, 1/29/08: Modeling with 1st Order Eqns., continued. Recommended HW: Sect 2.3, p. 64, #20,21,22,23,24,27,29 Challenge problem: p. 67, #32. Ask Isaac Newton or Jakob Bernoulli for the solution.
• Thurs, 1/31/08: Modeling, continued.
• Fri, 2/1/08: Autonomous Eqns and Population Dynamics. Recommended HW: Sect. 2.5, p. 88, #1,2,4,5,7,8,9,15
• Quiz 3: Fri, 2/1: Solution by variation of parameters and modeling. Click here to download solutions in PDF format.
• Mon, 2/4/08: Autonomous Eqns and Population Dynamics, continued. Recommended HW: Sect. 2.5, p. 89, #17,20,21,22,25,26,27,28 Note: #28 is relevant to chemical engineers.
• Tues, 2/5/08: Stability analysis for 1st order autonomous ODEs. Key concepts like equilibrium solution, asymptotic stability, unstable equilibrium are all defined in Sect. 2.5.
• Supplemental HW problem: Describe qualitative behavior of solutions to the ODE dy/dt = r(y-a)(y-b), where a is less than b and r is positive. Repeat for r negative.
• Thurs, 2/7/08: Review for Exam I
• Fri, 2/8/08: In-class Exam I. This will cover (i) analytic solution methods for 1st order ODEs (separation of variables, variation of parameters); (ii) modeling and applications; and (iii) qualitative analysis. Click here to download solutions in PDF format.
• Mon, 2/11/08: 2nd order linear ODE models---lecture combines parts of sect. 3.1 and sect. 3.8. HW: sect 3.1: #1,3,5,7.
• Tues, 2/12/08: General theory of nth order linear ODEs---lecture combines parts of sect. 3.2 and sect. 4.2. HW: sect 3.1: #9,10,13,15; Find general solution to ODE y''''-3y''+2y=0. Hint: Substitute s=r^2 into the characteristic, or auxiliary, equation.
• Thurs, 2/14/08: General theory, continued; Complex roots of characteristic equation. Click here to download notes on nth order linear homogeneous ODEs in PDF format. HW: sect 3.4: #1,3, 5,7.
• Fri, 2/15/08: Complex roots, continued. HW: sect 3.4, #11,17,19,21,29.
• Quiz 4: Fri, 2/15: General solution to 2nd order linear, homogeneous, constant coefficient ODEs in case (i) characteristic equation has distinct real roots; and (ii) characteristic equation has complex roots. Also, find solution given initial conditions. Click here to download solutions in PDF format.
• Mon, 2/18/08: Presidents' Day---No Class
• Tues, 2/19/08: Repeated roots. HW: sect 3.5, #3,5,7,11,12; Optional HW: #23,24,38.
• Thurs, 2/21/08: Method of Undetermined Coefficients. HW: sect 3.6, #1,3,7
• Fri, 2/22/08: Undetermined Coefficients, Continued. HW: sect 3.6, #13,15,17
• Quiz 5: Fri, 2/22: Solution to 2nd order linear, homogeneous, constant coefficient ODEs in case (i) characteristic equation has complex roots; (ii) characteristic equation has repeated roots. Click here to download solutions in PDF format.
• Mon, 2/25/08: Mechanical & Electrical Vibrations. HW: sect 3.8, #2,3,6,14,24
• Tues, 2/26/08: Mechanical & Electrical Vibrations, continued; Forced Vibrations. HW: sect 3.8, #12,18,20,26; sect. 3.9, #1,3,6,8
• Thurs, 2/28/08: Mechanical & Electrical Vibrations, continued.
• Fri, Leap Day, 2/29/08: Overview of Laplace Transforms (ch. 6). HW: sect 6.1, #5,6,7,9,15.
• Quiz 6: Fri, 2/29: Method of Undetermined Coefficients; Mechanical Systems; Vibrations. Click here to download solutions in PDF format.
• Mon, 3/3/08: Solution of IVPs using Laplace Transforms. HW: Sect 6.2, #1,3,5,11.
• Example of Laplace Transform Solution to ODE: Click here to download pdf file.
Useful link for partial fraction expansions: Wikibooks Calculus Integration Techniques--Partial Fractions
• Tues, 3/4/08: Solution of IVPs using Laplace Transforms, Continued. HW: Sect 6.2, #15,16,20,21
• Thurs, 3/6/08: Review for Exam II.
Exam II: Fri, 3/7 Open book; can use class handouts or material posted on the web (like quiz solutions). Exam will cover 2nd order linear ODEs, homogeneous solutions (real, repeated, and complex roots of characteristic equation), undetermined coefficient and Laplace transform solution techniques, and applications to mechanics and electrical circuits. Click here to download solutions in PDF format.
• Mon-Fri, 3/10/08-1/14/08: Spring Break---No Class
• Mon, 3/17/08: Step functions. HW: Sect 6.3, #1,2,4,7,9,14,17,18
• Tues, 3/18/08: ODEs with discontinuous forcing. HW: Sect 6.4, #1,3,5
• Thurs, 3/20/08: ODEs with discontinuous forcing, continued; Impulse functions. HW: Sect 6.4, #Solve the ODE y''+omega^2*y=f(t), y(0)=y'(0)=0, where the natural frequency omega=1 and the forcing function f(t) is the "square wave", given on in problem no. 27 on p. 331. The notation "omega^2" means the square, omega*omega. Is it possible to adjust the natuaral frequency so that resonance occurs with square wave forcing? HW: Sect 6.5, #1
• Fri, 3/21/08: No Class--University Day
• Practice Quiz (no grade assigned): Set up and solve the ODE for an electrical circuit with inductance L=1, resistance R=3, and capacitance C=1/2. Assume an AC (alternating current) source of the form cos(t) which is turned on at time t=1 and left on until time t=2, when it is turned back off, so the forcing voltage is E(t)=cos(t) for t between 1 and t and E(t)=0 for t less than 1 and for t bigger than 2. Also assume that at initial time t=0 there is no charge and no current in the circuit.
• Mon, 3/24/08: Impulse functions, continued. HW: Sect 6.5, #2,4,8,13
• Tues, 3/25/08: First order systems of ODEs. HW: Sect 7.1, #1,2,3,4,5,6,17
Useful links for linear algebra: http://ceee.rice.edu/Books/LA/index.html and http://en.wikibooks.org/wiki/Linear_algebra
• Thurs, 3/27/08: Conversion of higher order ODEs to 1st order system form; Linear system representation x'=Ax. HW: Sect 7.1, #18.
• Fri, 3/28/08: Review of matrices. HW: To be posted
• Quiz 7: Fri, 3/28: Impulse response functions, systems of ODEs. Be able to apply Laplace transforms to ODEs with terms involving Dirac delta and unit step functions. Also be able to convert ODEs to (possibly linear) system form. Click here to download solutions in PDF format.
• Mon, 3/31/08: Matrix operations (transpose, addition, scalar multiplication, matrix multiplication) HW from Rice University web page http://ceee.rice.edu/Books/LA/index.html: Sect 1.3 problems #1,2; Sect 2.3 problems #1acei,2,5,6; Sect 3.3 problems #2abcegm,3abc,6
• Tues, 4/1/08: Identity matrix, matrix inverse, linear systems of equations. HW from Rice U. web page: Sect 4.4 problems #3abcd,4a,5a,7,8
• Thurs, 4/3/08: Determinants. HW from Rice U. web page: Sect 5.7 problems #1abc,2ab,4b (assume 4a is true), 4c.
• Fri, 4/4/08: Fundamental Theorem of n x n linear systems, linear independence. HW to be posted. Click here to download supplemental notes on the Fundmental Theorem and Cramer's Rule for symbolically solving linear systems.
• Quiz 8: Fri, 4/4/08: Matrix operations (transpose, sum, scalar multiplication, matrix multiplication), Gauss-Jordan elimination, matrix inverses, determinants. Click here to download solutions in PDF format.
• Mon, 4/7/08: Eigenvalues and eigenvectors. HW from Rice University web page: Sect 9.1 problems #5,6,7.
• Tues, 4/8/08: Constant coefficient, homogeneous linear systems of ODEs (Sect. 7.5 in the textbook). HW: sect 7.5, #1,3,5,7, 15
• Thurs, 4/10/08: phase portraits, direction fields for 2 X 2 linear ODE systems (Sect. 7.5 in the textbook). HW: sect 7.5, #24,25,26;
• Fri, 4/10/08: Linear ODE systems with complex eigenvalues. HW: sect 7.6, #1,3,5. If possible, use software to compute the eigenvalues and eigenvectors.
• Quiz 9: Fri, 4/11/08: Eigenvalues and eigenvectors; solutions to 2 x 2 linear homogeneous ode systems. Click here to download solutions in PDF format.
• Mon, 4/14/08: Complex eigenvalues, continued. No new HW.
• Tues, 4/15/08: Repeated eigenvalues. HW: sect 7.8, #1,3,7,16.
• Thurs, 4/17/08: Method of variation of parameters for nonhomogeneous linear systems. Click here to download supplemental notes for this method. See also the supplemental notes from 4/4/08. HW. sect 3.7, #1,3,5. Convert 2nd order scalar ODE to 2 X 2 system and apply techniques from the supplemental notes.
• Fri, 4/18/08: Method of variation of parameters, continued.
• Quiz 10: Fri, 4/18/08: Solutions to ode systems with complex or repeated eigenvalues, phase portraits. Click here to download solutions in PDF format.
• Mon, 4/21/08: Review of linear algebra and linear systems of ODEs.
• Tues, 4/22/08: Optional Exam on linear algebra and linear systems of ODEs. If you choose to take this exam and your exam score is higher than your average quiz score (after dropping the 2 lowest quizzes), I'll replace your quiz grades with the exam grade. If you choose not to take the exam, or if your exam grade is lower than your quiz average, I'll not replace the quiz grades. Click here to download solutions in PDF format.
• Thurs, 4/24/08: Autonomous systems and stability. Classification of fixed points for linear systems. HW: sect 9.1, #1,3,5,7,17
• Fri, 4/25/08: Nonlinear systems of ODEs---some application examples. No quiz.
• Mon, 4/28/08: Stability, characterization of stability for linear systems in terms of eigenvalues, linearized stability analysis for nonlinear systems. Click here to download notes in PDF format.
• Tues, 4/29/08: Linearized stability analysis, continued. Sample Final Exam. Click here to download.
• M225 Common Hour Final Exam: Monday Night, May 5, 2008, from 6:00-7:50 pm. Section 01 Exam will take place in 103 Reid Hall
• Final Exam Solutions: Click here to download.