Time and Location

M W F Period 9 in Little Hall Room 113

Office Hours: M W F Period 8 in Little Hall Room 439

Textbook

Fundamentals of Differential Equations and Boundary Problems, Sixth Ed. by R. Kent Nagle, Edward B. Saff, and David Snider.

Description and Goals

Many problems, particularly those arising in science and engineering, are naturally stated in the language of differential equations. In this course, students will study solution techniques for ordinary differential equations (ODEs) and apply them to problems from the sciences.  Broadly speaking, we will develop ad hoc solutions for special types of first order ODEs and general solution techniques for higher order linear ODEs including the method of Laplace Transforms and power series methods.  This roughly corresponds to chapters 1,2,4,6,7, and 8 from the textbook with some omissions.

Homework

Suggested homework exercises will be given after covering each major topic and prior to each exam.  Answers will not be collected, but the problems will be representative of exam material.  Assignments will be repeated in this space when they are given in class.  Starred problems are more difficult. Mastery of them indicates a solid grasp of the material.

1/7 Class:  This was mainly terminology with a hint of things to come. Read 1.1,1.2.  Know what a DE is, order of a DE, ODE vs PDE, what a solution of a DE is, implicit vs. explicit solutions, solutions to IVPs.  Linear equations and existence/uniqueness (Theorem 1 on p. 11) will be covered in later classes, but it doesn’t hurt to read it now.  Exercises: (1.1) #1,3,5,7,13,15; (1.2) #1,3,5,7,9,11,17

1/9 Class: Existence and Uniqueness for 1st order ODEs, both in terms of definitions and geometrically as nonintersection of solution curves.  Direction Fields for qualitative analysis of solutions.  Use DField (http://math.rice.edu/~dfield/dfpp.html) to plot direction fields.  Exercises: (1.3) #1,3,5,7,9*

1/12 Class: Separable equations (Section 2.2).  Solve ODEs/IVPs via the method of separation of variables. Exercises: (2.2) #1-25 odd, 29*,30*,37

1/14 Class: Linear Equations (Section 2.3). Solve first order linear ODEs/IVPs.  Exercses: (2.3) #1-21 odd, 25a, 31*

1/16 Class: Exact Equations (Section 2.4).  Implicit differentiation (or use of multivariable chain rule) gives rise to exact equations.  Compatibility criterion to determine exactness.  Solution of exact equations.  Exercises: (2.4): #1-27 odd,  29*,31*32*

1/21 Class: Integrating Factors to make ODEs exact (Section 2.5).  Know the general notion of an integrating factor.  Find integrating factors when they are a function of only x or only y.  Exercises: (2.5) #1-13 odd, 15*,16*,17*

1/23 Class: Substitutions/Change of Variables (Section 2.6).  Be able to perform changes of variables.  Specific types:  Homogeneous equations and Bernoulli equations.  Exercises: (2.6): #1-7 odd (ignore non-Bernoulli/Homogenous), 9-15 odd, 21-27 odd, 41,42,47ab*

1/25 Class: Mathematical Modeling (Section 3.1) and Newtonian Mechanics (Section 3.4). The overall mathematical modeling endeavor.  Solving problems of Newtonian mechanics by turning F=ma into a differential equation.  Exercises: Read 3.1, (3.4) #1,5 (Just find the equation of motion for these), 15,19

Sample Exam I: Exam (Solutions). You should attempt the practice exam after doing your usual studying. Since the questions are taken randomly from the testable material, you should not focus too much on the particular problems asked on this sample exam.

1/28 Class: Mass-Spring Systems (Section 4.1).  2nd order constant coefficient linear equations.  Using the Characteristic/Auxilary Polynomial to find solutions to constant coefficient equations. Exercises: Read 4.1, (4,1): #3,5,6,7; (4.2): #21

2/2 Class: General solutions to constant coefficient equations (Section 4.2).  Linear combinations of solutions and linear (in)dependence of solutions.  Existence and uniqueness for 2nd order constant coefficient linear ODEs.  General solution is generated by linear combinations of two linearly independent solutions.  Exercises: (4.2): #27,29,31,34* (if you know what a determinant is),35,36,37,39.

2/4 Class: General solutions to constant coefficient equations (Sections 4.2,4.3).  We now know how to find the general solution to any constant coefficient equation/solve any associated IVP.  Exercises: (4.2): 1-19 odd, 37-41 odd, 42*, 43, (4.3): 1-27 odd, 28, 29

2/6 Class: The nonhomogeneous case (some 4.4, mostly 4.5).  Vector Spaces and Linear Operators (Key example: Twice differentiable functions as a vector space, 2nd order linear ODE as an operator).  The superposition principle (p.182 Thm 3).  Exercises: Enrich yourselves by reading the wikipedia page on vector spaces.

2/9 Class: Methods of finding solutions in the nonhomogeneous case (4.4). Solution by factoring the equation in differential operator form (I will write up some notes on this since it does not appear in our textbook).  Solution by the method of undetermined coefficients.  Exercises: (4.4) 1-35 odd (just work enough to understand the method).  Try to solve a few by factoring the ODE.  Compare your answers.

2/11 Class: Method of Undetermined Coefficients reviewed (Section 4.4, some 4.5).  Particular solutions to ODEs and solutions to IVPs using undetermined coefficients.  Variation of Parameters (Section 4.6).  Exercises: (4.5) #1,2,3-15 odd, 17,19, 23,25,31-39 odd, 41*, 48

2/13 Class: Variation of Parameters (Section 4.6).  Exercises: (4.6) #1-17 odd

2/16 Class: Variable Coefficients (4.7).  Almost Everything still works (restricted to intervals of continuity).  Notable exceptions:  No way to find the general homogeneous solution, no method of undetermined coefficients.  Reduction of Order to find a second linearly independent homogeneous solution.  Exercises: (4.7) #1-7 odd, 27*, 29*,31,32*,34,35,36,37,39,45,47,51*

2/18  Class: Cauchy-Euler Equations (4.7). General solutions to homogeneous Cauchy-Euler.  Use the homogeneous solution with variation of parameters to find general solutions to nonhomogeneous equations.  Exercises: (4.7) #9-17 odd, 37,39,41,43

2/20,2/23 Class: Free Mechanical Vibration (4.9). Representation of a mass-spring system as an ODE/IVP.  The relationship between the characteristic polynomial and under/over/critically damped systems.  Exercises: (4.9) #1,3,7,9,13*15*,16*,17*

Sample Exam II: Exam (Solutions)

2/25 Class: Exam II

2/27 Class: Review of what went wrong on Exam II and your feedback.

3/9 Class: Introduction to the Laplace Transform.  We developed the Laplace transform as a generalization of power series, resulting in the definition of the transform.  We then sketched an outline of how we will use the Laplace transform to solve ODEs and computed a simple transform.  Exercises: (7.2) #1-12 (These are just solving improper integrals)

3/11 Class: Properties of the Laplace Transform (7.2).  We computed the transforms of sine and cosine.  Next, we established the linearity of the transform.  Finally, we gave two sufficient conditions on a function to ensure the existence of its Laplace transform: Piecewise Continuity and Exponential Order. Exercises (7.2) #13,15,21,23,25,28,29,30*,31*,32,33*.

3/13 Class: More properties of the Laplace Transform (7.3).  Shifting by multiplication by exponentials. Laplace transforms of derivatives.  Derivatives of Laplace transforms.  Exercises: (7.3) #1-9 odd, 13,15 (remember power reduction) 25,27*,29*,37*.

3/16 Class: Inverse Laplace Transform I (7.4).  Definition of the inverse transform.  Linearity of the inverse.  Examples of taking the inverse transform using algebraic manipulation.  We can’t do many problems yet, but you should be able to do (7.4) #1-10.

3/18 Class: Inverse Laplace Transform II (7.4).  We can take the transform of any rational function using partial fractions so long as no repeated quadratic factors appear in the denominator.  Exercises: (7.4) #21-29 odd (just choose a few of these), 31,33,35.

3/20 Class: Solving ODEs with the Laplace Transform (7.5).  Solutions to 2nd order constant coefficient IVPs.  Shifting the initial conditions when needed.  Exercises: (7.5) A few from #1-9 odd, 11,13,15,17,23,24,25,29,33,34.

3/23 Class: Solving ODEs with the Laplace Transform (7.5). Solutions to (some) variable coefficient IVPs.  Asymptotic behavior of the Laplace transform for PWC, exponential order functions.  Exercises: (7.5) #35,36,37,38.

3/25 Class: (Inverse) Transforms of periodic and discontinuous functions (7.6).  We picked up a few formulas for transforming and inverse transforming discontinuous functions.  We also discussed transforms of periodic functions.  Exercises: (7.6) #1-4,5,7,9, a few of 11-18, 19,21,23,25,27, a few of 29-32, a few of 33-38,39,43*

3/27 Class: Convolutions (7.7).  We mainly worked examples from 7.6, but we did discuss the definition of the convolution and how convolutions “smooth” out functions.  We stated the convolution theorem for Laplace transform and hastily showed how we can use it to handle the repeated quadratic factors from partial fractions.  Exercises: (7.7) #1-4,5,7,13,31,35*

3/30 Class: The Dirac Delta (7.8). Physicial interpretation of the delta function (which isn’t actually a function).  Laplace transform of the delta function.  Solving ODEs involving the delta function.  Exercises: (7.8) #1,3,7,9,11, a few from 13-14, 21

Sample Exam III (Solutions) .  This is longer than what I’d do in reality, but even so the sample does not represent all of the topics that may be covered.

4/6 Class: Power Series Solutions (8.1).  Use IVPs to construct Taylor polynomials to approximate a solution.  This cannot always be done.  Exercises: (8.1) #1-7 odd, 9,11*,12*,13.  You should also review the material of 8.2 as a review of what you should already know.  The problems in (8.2) with few exceptions should seem doable.

4/8 Class: Power Series Solutions to Linear Equations (8.3).  Definitions of analytic (at a point), ordinary point, singular point.  Finding solutions at ordinary points.  Exercises: (8.3) #1-7 odd, 11,13,18,19,21,25,27,32*,33*

4/10 Class: We spent most of class working examples of 8.3 material.  Same exercises as before.

4/13 Class: A mixture of 8.3 and 8.4.  Lower bounds on the radius of convergence for a power series solution.  Shifting ODEs to in turn shift the center of our power series expansion to x=0 to simplify computations.  Exercises: (8.4) #1-11 odd,13,15,17,21 (look at #20 and the discussion at the very end of the section’s text)

4/15 Class: More 8.4.  We looked at the solutions to nonhomogeneous linear equations.  Exercises: (8.4) #23-27 odd. These are the same type of problem as #21 from the previous class’s exercises.

To study for Exam IV: Do one problem from each of the following sections:

• 7.8 #13-19 odd
• 8.1 # 1-9 odd
• 8.3 # 7-11 odd
• 8.3 # 1-9 odd
• 8.3 # 11-17 odd (also find recurrence relation)
• 8.4 # 21-27 odd

If you answered the above all correctly, then you are prepared for the exam.

Final Exam Review.  Please look at Sakai.

 

Quizzes

Occasionally there will be unannounced quizzes which will consist of a single, simple problem from recently covered material.  Quizzes count for a very small portion (4%) of the overall grade, but poor quiz scores should be taken as a sign to get assistance during office hours.  There is no makeup opportunity if quizzes are missed, but the lowest few quiz scores will be omitted.

Final Grades

96% of your grade will come from four in-class exams (24% per exam).  The final 4% comes from quizzes.  A final exam may be taken to replace the score of one of the four exams.

Exam Schedule

Exam I (Exam,Solutions): Friday January 30 during normal class time

Exam II (Exam, Solutions): Wednesday February 25th during normal class time

Exam III (Exam, Solutions): Friday April 3 during normal class time

Exam IV (Exam, Solutions): Wednesday April 22 during normal class time

Final Exam: May 1, 10:00-12:00 PM in LIT 113 (Our usual room)

The four regular exams will occur during normal class times.  As we progress through the term, dates will appear here.

Grading Scale

A: 90-100, B:80-89, C: 70-79, D: 60-69 with the top and bottom two points from each reserved for plus and minus grades.

Course Policies

• Calculators or other electronic devices are prohibited during in class quizzes or exams.
• While attendance is not mandatory, there are absolutely no exam makeups without verifiable documentation.  There are no quiz makeups at all.
• If you are requesting disability assistance, you must first register with the Dean of Students office (https://www.dso.ufl.edu/drc).  They will provide you the proper documentation to present to me.