{"id":219,"date":"2015-01-04T00:33:21","date_gmt":"2015-01-04T05:33:21","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/rptacek\/?page_id=219"},"modified":"2026-03-19T08:23:29","modified_gmt":"2026-03-19T12:23:29","slug":"map-2302-spr15","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/rptacek\/previous-classes\/map-2302-spr15\/","title":{"rendered":"MAP 2302-3731 (Spr 15)"},"content":{"rendered":"\r\n
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MAP 2302-3731 (Spr 15)<\/h1>\n\n\n\n

Time and Location<\/h3>\n\n\n\n

M W F Period 9\u00a0in Little Hall Room\u00a0113<\/p>\n\n\n\n\n\n

Office Hours: M W F Period 8\u00a0in Little Hall Room 439<\/p>\n\n\n\n

Textbook<\/h3>\n\n\n\n

Fundamentals of Differential Equations and Boundary Problems, Sixth Ed. by R. Kent Nagle, Edward B. Saff, and David Snider.<\/p>\n\n\n\n

Description and Goals<\/h3>\n\n\n\n

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. \u00a0Broadly speaking, we will develop ad hoc solutions for\u00a0special types of first order ODEs\u00a0and general solution techniques for higher order linear\u00a0ODEs including the method of Laplace Transforms and power series methods. \u00a0This roughly corresponds to chapters 1,2,4,6,7, and 8 from the textbook with some omissions.<\/p>\n\n\n\n

Homework<\/h3>\n\n\n\n

Suggested homework exercises will be given after covering each major topic and prior to each exam. \u00a0Answers will not be collected, but\u00a0the problems will be representative of exam material. \u00a0Assignments will be repeated in this space when\u00a0they are given in class. \u00a0Starred problems are more difficult. Mastery of them indicates a solid\u00a0grasp of the material.<\/p>\n\n\n\n\n\n

1\/7 Class<\/strong>: \u00a0This was mainly terminology with a hint of things to come. Read 1.1,1.2<\/strong>. \u00a0Know what a DE is, order of a DE, ODE vs PDE, what a solution of a DE is, implicit vs.\u00a0explicit solutions, solutions to IVPs. \u00a0Linear equations<\/strong> and existence\/uniqueness<\/strong> (Theorem 1 on p. 11) will be covered in later classes, but it doesn’t hurt to read it now. \u00a0Exercises: (1.1) #1,3,5,7,13,15; (1.2) #1,3,5,7,9,11,17<\/strong><\/p>\n\n\n\n\n\n

1\/9 Class:\u00a0<\/strong>Existence and Uniqueness for 1st order ODEs, both in terms of definitions and geometrically as nonintersection of solution curves. \u00a0Direction Fields for qualitative analysis of solutions. \u00a0Use DField (http:\/\/math.rice.edu\/~dfield\/dfpp.html<\/a>) to plot direction fields. \u00a0Exercises: (1.3) #1,3,5,7,9*<\/strong><\/p>\n\n\n\n\n\n

1\/12 Class:\u00a0<\/strong>Separable equations (Section 2.2). \u00a0Solve ODEs\/IVPs via the method of separation of variables. Exercises: (2.2) #1-25 odd, 29*,30*,37<\/strong><\/p>\n\n\n\n\n\n

1\/14 Class:<\/strong> Linear Equations (Section 2.3). Solve first order linear ODEs\/IVPs. \u00a0Exercses: (2.3) #1-21 odd, 25a, 31*<\/strong><\/p>\n\n\n\n\n\n

1\/16 Class:<\/strong> Exact Equations (Section 2.4). \u00a0Implicit differentiation (or use of multivariable chain rule) gives rise to exact equations. \u00a0Compatibility criterion to determine exactness. \u00a0Solution of exact equations. \u00a0Exercises: (2.4): #1-27 odd, \u00a029*,31*32*<\/strong><\/p>\n\n\n\n\n\n

1\/21 Class:<\/strong> Integrating Factors to make ODEs exact (Section 2.5). \u00a0Know the general notion of an integrating factor. \u00a0Find integrating factors when they are a function of only x or only y. \u00a0Exercises: (2.5) #1-13 odd, 15*,16*,17*<\/strong><\/p>\n\n\n\n\n\n

1\/23 Class:<\/strong> Substitutions\/Change of Variables (Section 2.6). \u00a0Be able to perform changes of variables. \u00a0Specific types: \u00a0Homogeneous equations and Bernoulli equations. \u00a0Exercises: (2.6): #1-7 odd (ignore non-Bernoulli\/Homogenous), 9-15 odd, 21-27 odd, 41,42,47ab*<\/strong><\/p>\n\n\n\n\n\n

1\/25 Class:\u00a0<\/strong>Mathematical Modeling (Section 3.1) and Newtonian Mechanics (Section 3.4). The overall mathematical modeling endeavor. \u00a0Solving problems of Newtonian mechanics by turning F=ma into a differential equation. \u00a0Exercises: Read 3.1, (3.4) #1,5 (Just find the equation of motion for these), 15,19<\/strong><\/p>\n\n\n\n\n\n

Sample\u00a0Exam I:\u00a0<\/strong>Exam<\/a>\u00a0(Solutions<\/a>). 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.<\/p>\n\n\n\n\n\n

1\/28 Class:\u00a0<\/strong>Mass-Spring Systems (Section 4.1). \u00a02nd order constant coefficient linear equations. \u00a0Using the Characteristic\/Auxilary Polynomial to find solutions to constant coefficient equations. Exercises: Read 4.1, (4,1): #3,5,6,7; (4.2): #21<\/strong><\/p>\n\n\n\n\n\n

2\/2 Class:\u00a0<\/strong>General solutions to constant coefficient equations (Section 4.2). \u00a0Linear combinations of solutions and linear (in)dependence of solutions. \u00a0Existence and uniqueness for 2nd order constant coefficient linear ODEs. \u00a0General solution is generated by linear combinations of two linearly independent solutions. \u00a0Exercises: (4.2): #27,29,31,34* (if you know what a determinant is),35,36,37,39.<\/strong><\/p>\n\n\n\n\n\n

2\/4 Class:<\/strong> General solutions to constant coefficient equations (Sections 4.2,4.3). \u00a0We now know how to find the general solution to any constant coefficient equation\/solve any associated IVP. \u00a0Exercises: (4.2): 1-19 odd, 37-41 odd, 42*, 43, (4.3): 1-27 odd, 28, 29<\/strong><\/p>\n\n\n\n\n\n

2\/6 Class:\u00a0<\/strong>The nonhomogeneous case (some 4.4,\u00a0mostly\u00a04.5). \u00a0Vector Spaces and Linear Operators (Key example: Twice differentiable functions as a vector space, 2nd order linear ODE as an operator). \u00a0The superposition principle (p.182 Thm 3). \u00a0Exercises: Enrich yourselves by reading the wikipedia page on vector spaces.<\/strong><\/p>\n\n\n\n\n\n

2\/9 Class:<\/strong> 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). \u00a0Solution by the method of undetermined coefficients. \u00a0Exercises: (4.4) 1-35 odd (just work enough to understand the method). \u00a0Try to solve a few by factoring the ODE. \u00a0Compare your answers.<\/strong><\/p>\n\n\n\n\n\n

2\/11 Class:<\/strong> Method of Undetermined Coefficients reviewed (Section 4.4, some 4.5). \u00a0Particular solutions to ODEs and solutions to IVPs using undetermined coefficients. \u00a0Variation of Parameters (Section 4.6). \u00a0Exercises: (4.5) #1,2,3-15 odd, 17,19, 23,25,31-39 odd, 41*, 48<\/strong><\/p>\n\n\n\n\n\n

2\/13 Class:<\/strong> Variation of Parameters (Section 4.6). \u00a0Exercises: (4.6) #1-17 odd<\/strong><\/p>\n\n\n\n\n\n

2\/16 Class:<\/strong> Variable Coefficients (4.7). \u00a0Almost Everything still works (restricted to intervals of continuity). \u00a0Notable exceptions: \u00a0No way to find the general homogeneous solution, no method of undetermined coefficients. \u00a0Reduction of Order to find a second linearly independent homogeneous solution. \u00a0Exercises: (4.7) #1-7 odd, 27*, 29*,31,32*,34,35,36,37,39,45,47,51*<\/strong><\/p>\n\n\n\n\n\n

2\/18 \u00a0Class:\u00a0<\/strong>Cauchy-Euler Equations (4.7). General solutions to homogeneous Cauchy-Euler. \u00a0Use the homogeneous solution with variation of parameters to find general solutions to nonhomogeneous equations. \u00a0Exercises: (4.7) #9-17 odd, 37,39,41,43<\/strong><\/p>\n\n\n\n\n\n

2\/20,2\/23 Class:<\/strong> Free Mechanical Vibration (4.9). Representation of a mass-spring system as an ODE\/IVP. \u00a0The relationship between the characteristic polynomial and under\/over\/critically damped systems. \u00a0Exercises: (4.9) #1,3,7,9,13*15*,16*,17*<\/strong><\/p>\n\n\n\n\n\n

Sample Exam II:<\/strong>\u00a0Exam<\/a> (Solutions<\/a>)<\/p>\n\n\n\n\n\n

2\/25 Class: Exam II<\/strong><\/p>\n\n\n\n\n\n

2\/27 Class:\u00a0<\/strong>Review of what went wrong on Exam II and your feedback.<\/p>\n\n\n\n\n\n

3\/9 Class:\u00a0<\/strong>Introduction to the Laplace Transform. \u00a0We developed the Laplace transform as a generalization of power series, resulting in the definition of the transform. \u00a0We then sketched an outline of how we will use the Laplace transform to solve ODEs and computed a simple transform. \u00a0Exercises: (7.2) #1-12<\/strong> (These are just solving improper integrals)<\/p>\n\n\n\n\n\n

3\/11 Class:\u00a0<\/strong>Properties of the Laplace Transform (7.2). \u00a0We computed the transforms of sine and cosine. \u00a0Next, we established the linearity of the transform. \u00a0Finally, 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*.<\/strong><\/p>\n\n\n\n\n\n

3\/13 Class:\u00a0<\/strong>More properties of the Laplace Transform (7.3). \u00a0Shifting by multiplication by exponentials. Laplace transforms of derivatives. \u00a0Derivatives of Laplace transforms. \u00a0Exercises: (7.3) #1-9 odd, 13,15 (remember power reduction) 25,27*,29*,37*.<\/strong><\/p>\n\n\n\n\n\n

3\/16 Class:<\/strong> Inverse Laplace Transform I (7.4). \u00a0Definition of the inverse transform. \u00a0Linearity of the inverse. \u00a0Examples of taking the inverse transform using algebraic manipulation. \u00a0We can’t do many problems yet, but you should be able to do (7.4) #1-10.<\/strong><\/p>\n\n\n\n\n\n

3\/18 Class:<\/strong> Inverse Laplace Transform II (7.4). \u00a0We can take the transform of any rational function using partial fractions so long as no repeated quadratic factors appear in the denominator. \u00a0Exercises: (7.4) #21-29 odd (just choose a few of these), 31,33,35.<\/strong><\/p>\n\n\n\n\n\n

3\/20 Class:<\/strong> Solving ODEs with the Laplace Transform (7.5). \u00a0Solutions to 2nd order constant coefficient IVPs. \u00a0Shifting the initial conditions when needed. \u00a0Exercises: (7.5) A few from #1-9 odd, 11,13,15,17,23,24,25,29,33,34.<\/strong><\/p>\n\n\n\n\n\n

3\/23 Class:<\/strong> Solving ODEs with the Laplace Transform (7.5). Solutions to (some) variable coefficient IVPs. \u00a0Asymptotic behavior of the Laplace transform for PWC, exponential order functions. \u00a0Exercises: (7.5) #35,36,37,38.<\/strong><\/p>\n\n\n\n\n\n

3\/25 Class: <\/strong>(Inverse)\u00a0Transforms of periodic and discontinuous functions (7.6). \u00a0We picked up a few formulas for transforming and inverse transforming discontinuous functions. \u00a0We also discussed transforms of periodic functions. \u00a0Exercises: (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*<\/strong><\/p>\n\n\n\n\n\n

3\/27 Class:<\/strong> Convolutions (7.7). \u00a0We mainly worked examples from 7.6, but we did discuss the definition of the convolution and how convolutions “smooth” out functions. \u00a0We stated the convolution theorem for Laplace transform and hastily showed how we can use it to handle the repeated quadratic factors from partial fractions. \u00a0Exercises: (7.7) #1-4,5,7,13,31,35*<\/strong><\/p>\n\n\n\n\n\n

3\/30 Class:<\/strong> The Dirac Delta (7.8). Physicial interpretation of the delta function (which isn’t actually a function). \u00a0Laplace transform of the delta function. \u00a0Solving ODEs involving the delta function. \u00a0Exercises: (7.8) #1,3,7,9,11, a few\u00a0from 13-14, 21<\/strong><\/p>\n\n\n\n\n\n

Sample Exam III<\/a>\u00a0(Solutions<\/a>)\u00a0. \u00a0This is longer than what I’d do in reality, but even so the sample does not<\/strong> represent all of the topics that may be covered.<\/p>\n\n\n\n\n\n

4\/6 Class:\u00a0<\/strong>Power Series Solutions (8.1). \u00a0Use IVPs to construct Taylor polynomials to approximate a solution. \u00a0This cannot always be done. \u00a0Exercises: (8.1) #1-7 odd, 9,11*,12*,13. \u00a0<\/strong>You should also review the material of 8.2 as a review of what you should already know. \u00a0The problems in (8.2)\u00a0<\/strong>with few exceptions should seem doable.<\/p>\n\n\n\n\n\n

4\/8 Class:<\/strong> Power Series Solutions to Linear Equations (8.3). \u00a0Definitions of analytic (at a point), ordinary point, singular point. \u00a0Finding solutions at ordinary points. \u00a0Exercises: (8.3) #1-7 odd, 11,13,18,19,21,25,27,32*,33*<\/strong><\/p>\n\n\n\n\n\n

4\/10 Class:<\/strong>\u00a0We spent most of class working examples of 8.3 material. \u00a0Same exercises as before.<\/p>\n\n\n\n\n\n

4\/13 Class:<\/strong> A mixture of 8.3 and 8.4. \u00a0Lower bounds on the radius of convergence for a power series solution. \u00a0Shifting ODEs to in turn shift the center of our power series expansion to x=0 to simplify computations. \u00a0Exercises: (8.4) #1-11 odd,13,15,17,21<\/strong> (look at #20 and the discussion at the very end of the section’s text)<\/p>\n\n\n\n\n\n

4\/15 Class:<\/strong> More 8.4. \u00a0We looked at the solutions to nonhomogeneous linear equations. \u00a0Exercises: (8.4) #23-27 odd.\u00a0<\/strong>These are the same type of problem as #21 from the previous class’s exercises.<\/p>\n\n\n\n\n\n

To study for Exam IV:\u00a0<\/strong>Do one problem from each of the following sections:<\/p>\n\n\n\n