 ## Math 284 Differential Equations (Spring 2020)

 Instructor Mark Pernarowski Textbook Differential Equations (9th ed.), Nagle, Saff, Snider Section 01 Office Hours Schedule (Wil 2-236) Phone 994-5356 Classroom MTWF 11:00-11:50am (Wil 1-142) Grading: The course % is determined by:   Midterm 1      M1           100    Midterm 2      M2           100   Final                F            100   Quizzes           Q           100  ________________________________                                        400          % = (M1+M2+F+HW)/4  The final is not comprehensive. Six quizzes each worth 20 points will be given. Your best 5 quiz scores determine Q above. Exam and quiz dates are indicated on the schedule below. Their content will be announced in class.All exams and quizzes are closed book and no electronic devices are permitted. This includes phones!! Syllabus: Material covered in text is from:Chapter 1 Introductory Definitions Chapter 2 First Order ODE Methods Chapter 3 First Order Models Chapter 4 Second Order Linear ODE Methods Chapter 6 Higher Order Differential Equations Chapter 7 Laplace Transforms Chapter 9 Linear Systems   Homework: Suggested homework is listed below. Although the homework is not graded it is representative of the kinds of questions which will be on quizzes and exams. Some additional problem sets and/or handouts will be handed out in class and/or posted on this site below. Schedule
Below is a calendar showing the schedule of quizzes (orange) and tests (red) and  holidays (green).

 Sunday Monday Tuesday Wed Thursday Friday Saturday 12 13       (1.1)Classes start 14        (1.2) 15.       (1.2)/(2.1) 16 17        (2.2) 18 19 20MLK Day 21        (2.2)/(2.3) 22        (2.3)/(2.4) 23 24.       (2.4)Quiz 1 25 26 27       (2.6) 28        (2.6) 29        (2.6) 30 31        (3.2) 1 2 3.        (3.2) 4          (3.4) 5         (3.5) 6 7.       (4.1)Quiz 2 8 9 10       (4.2) 11.       (4.3) 12Review 13 14 Midterm 1 15 16 17 Pres. Day 18       (4.2) 19      (4.2) 20 21       (4.4) 22 23 24       (4.4) 25    (4.5)/(4.6) 26     (4.6) 27 28       (4.7)Quiz 3 29 1 2         (4.7) 3         (4.7) 4       (4.9) 5 6  (4.9)/(4.10) 7 8 9        (7.2) 10      (7.2) 11 Quiz 4 12 13      (7.3) 14 15 16Spring Break 17Spring Break 18Spring Break 19Spring Break 20Spring Break 21 22 23 24 Review 25Midterm 2 26NCUR 27 NCUR 28 29 30 31 1 2 3 4 5 6 7 8Quiz 5 9 10Univer. Day 11 12 13 14 15 16 17 18 19 20 21 22 23 24Quiz 6 25 26 27 28 29 Review 30 1 Classes EndReview 2 3 4 5 6 7  Final6-7:50pm 8 9 ## Suggested Homework and Syllabus

 1.1 1,2,5,7,9,11 Dependent/independent variables, linear ODE 1.2 1a,2a,3,5,7,9,11,21,23,27,29a Solutions, Existence, Initial Value Problem 1.3 not covered Direction Fields 1.4 not covered Euler's Method 2.1 none Motion of a Falling Body 2.2 1,2,5,7,9,11,17,18,19,23,25,27,35,37 1rst Order Separable 2.3 2,3,4,7,9,13,15,17,18,19,22,37 1rst Order Linear 2.4 1,3,5 (solve as well),11,13,22,25,26 1rst Order Exact 2.5 not covered 1rst Order Special Integrating Factors 2.6 5,7,9,11 (implicit),15,21,23,25 1rst Order Homogeneous and Bernoulli only Suggested HW below is still tentative. 3.1 none Mathematical Modelling 3.2 1,3,7 Mixing models (only) 3.3 not covered Heating and Cooling Problems 3.4 1,5,24(hard) Newtonian Mechanics 3.5 not covered Electrical Circuits 3.6 not covered Improved Euler Methods 3.7 not covered Higher Order Numerical Methods Midterm 1 Content Summary Below REVIEW MATERIAL WILL BE EMAILED FOR MIDTERM 1 4.1 none Introductory 2nd Order Models 4.2 1,5,9,13,19,27,31,37(r=1 root), 39 (r=2), 43 Homogeneous IVP, existence, Real Roots Case 4.3 1,3,5,9,11,13,19(r=1),21,25,29b (r=2),29c Homogenous, Complex Roots Case 4.4 9,11,13,15,17,23,25 (ugly),33 Nonhomogeneous: Undetermined Coeff. 4.5 3,7,17,19,23,25,27,33 (trig ident for cos^3),35 Nonhomogeneous: General solutions 4.6 1,3,5,7,11,13,17(longish) Variation of Parameters 4.7 9,11,13,15,17,19, Reduction of Order: 41,43,45 Cauchy-Euler equations, Reduction of Order 4.8 not covered Qualitative theory 4.9 1,7,9,11 Mechanical Vibrations 4.10 Covered but not on exam Mechanical Vibrations: Forced Midterm 2 Chapter 4 on HW material assigned REVIEW MATERIAL WILL BE EMAILED 5 Time permitting at end of course Phase Plane, Numerical 6 Time permitting at end of course General Theory of Linear Equations 7.2 3,5,9,11,13,15,17 Laplace Transform Definition 7.3 1,3,5,7,9,13,25,31 Laplace Transform Properties 7.4 1,3,7,9,21,23,25 (last 3 are nastier),33,35 Laplace Transform Inverse 7.5 1,3,7(nasty),11 (set y(t)=w(t-2)),15,17,19,35 Laplace Transform Initial Value Problems 7.6 not covered Laplace Transform Discontinuous Functions 7.7 not covered Laplace Transform of Periodic Functions 7.8 1,2,3,5,7,9,13 Laplace Transform Convolution Theorem 7.9 not covered Laplace Transform - delta function 7.10 not covered Laplace Transform - Systems of Equations 8 not covered Series Approximations and Solutions 9.1 1,3,5,8,11 Differential Equations as Systems 9.2 none Linear Algebraic Equations Gaussian Elimination 9.3 1,3,5,7b,7c,8,9,17,21,27,31,33,35,37,39 Matrix algebra and Calculus 9.4 1,3,5,9,13,15,19, 28!! Linear Systems - Normal Form 9.5 1,3,5,7,11,19,21,31!! Linear Systems - Constant Coefficient (Real Case) 9.6 1, 3 (given lamba=1),5,13a Linear Systems - Constant Coefficient (Complex Case) 9.7 11,13,21a Linear Systems - Variation of Parameters 9.8 Review problems - WILL BE EMAILED Linear Systems - Generalized eigenvectors and Repeated eigenvalues. Final Chapter 7 and 9 REVIEW MATERIAL WILL BE EMAILED Review questions: Laplace Transforms  and Systems REVIEW MATERIAL WILL BE EMAILED

## Exam and Quiz Outlines

Quizzes

 Quiz 1 1.1,1.2,2.2 Basic definitions (linear, order,......), explicit and implicit differential equations, verifying y(x) is a solution, finding ODE for implicit solutions, separable equations and solving Initial Value Problems (IVP). There will NOT be anything on the falling body problem in (2.1). Quiz 2 2.3,2.4,2.5 Solving Linear, Exact, Homogenous and Bernoulli Equations. Will be required to solve at least three of these four equations. Also, there will be one "matching" problem where you'll need to identify if the given eqn(s) is Linear, Separable, Exact, Homogenous  and/or Bernoulli. Quiz 3 4.2-4.5 Constant coefficient second order general solutions: 3 cases: real distinct, real repeated and complex. Solving Initial Value Problems. Method of undetermined coefficients for all cases of forms f(x)=P_(x), P_n(x)e^ax, P_n(x)* trig. Could be asked to find particular soln y_p(x) and/or the appropriate form of y_p(x) especially when f(x) contains the homogeneous solution. There will NOT be any questions on higher order (3rd, 4rth) problems. What we did in class was part of section 6.2. Quiz 4 4.6,4.7 Variation of Parameters, Cauchy-Euler equations (homogenous and nonhomogenous), Reduction of order, Quiz 5 TBA Quiz 6 TBA

Midterm 1

The exam will cover material from the following sections of the textbook:

1. Section 1.1  ODE definitions and theory
2. Section 1.2  IVP explicit/implicit solutions, existence uniqueness
3. Section 2.2  Separable Equations
4. Section 2.3  Linear Equations
5. Section 2.4  Exact Equations
6. Section 2.6  Homogeneous Equations and Bernoulli Equations only
7. Section 3.2  Mixing Problems (no population problems)
8. Section 3.4  Newtonian Mechanics - falling bodies, friction, rockets

Notes:

• You will have to solve a separable, linear, exact, homogeneous and Bernoulli equation. This forms the bulk of the exam ( about 70%)
• There will be one application problem, either a mixing problem or a rocket problem (15%)
• One question will require you to categorize types of differential equations (15%).
• The sample problems EMAILED are a good indication of the difficulty level of the problems.
• Change: there will be no Newtonian Mechanic problem!! ......only a mixing problem (item 7 above)

Midterm 2

The exam will cover material from the following sections of the textbook: 4.2-4.7, 4.9

1. Constant Coefficient 2nd order homogeneous yh(t)
2. Constant Coefficient 3rd order homogeneous yh(t) with one solution known (see review sheet)
3. Constant Coefficient 2nd order: Undetermined Coefficients Method for yp(t)
4. General Solutions y(t)=yh(t) + yp(t), Initial Value Problems, Wronskian for independence
5. Cauchy Euler 2nd Order homogeneous yh(t)
6. Variation of Parameter Method for yp(t) - standard form.
7. Reduction of order: homogeneous solution y2(t) from given homogeneous y1(t)
8. Mechanical Vibrations: Amplitude Phase Form y= A sin(wt+phi) for no friction case

Notes:

• There will be an amplitude-phase problem (10-15%). In fact, there will be a question from each point 1-8 above with the sole possible exception of 2.
• The sample review problems EMAILED 3/11 are a good indication of the difficulty level of the problems but this  sheet has only one amplitude-phase problem.
• Undetermined coefficients is ONLY for L(y)=ay''+by'+cy=f and not L(y)=ax2y''+bxy'+cy=f

### Final (Tentative)

TBA, Wil 1-142

1. Sections of the textbook covered: 7.2-7.5, 7.7, 9.4-9.7 and repeated eigenvalues notes
2. Sample Problems: Laplace Transforms (Ch 7)  and Systems (Ch 9)
3. Laplace Transform Table is attached to test. DO NOT BRING YOUR OWN!!
4. The exam will be about 60% on systems and about 40% on Laplace transforms

#### Topics Covered

• Laplace: using definition to calculate F(s) for discontinuous functions
• Laplace: Taking transforms using tables and properties
• Laplace: Inversion via partial fractions and completing the square
• Laplace: Solving Initial Value Problems
• Laplace: Using convolution theorem to solve IVP and invert transforms
• Systems: Matrix inverse (2x2) and basic matrix calculus: (AX)'=AX'+A'X
• Systems: Independence, Wronskian, Fundamental Matrix X(t)
• Systems: General Solution for homogeneous/nonhomogeneous systems
• Systems: Solving Initial Value Problems using fundamental matrix X(t)
• Systems: Constant A (2x2): real distinct eigenvalues
• Systems: Constant A (2x2): real repeated eigenvalues
• Systems: Constant A (2x2): complex eigenvalue
• Systems: Variation of Parameters

Updated on: 12/20/2019.