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     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)

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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.





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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
 
20
MLK 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)

12
Review
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
 
16
Spring Break
17
Spring Break
18
Spring Break
19
Spring Break
20
Spring Break
21
 
22
 
23 24
Review
25
Midterm 2
26
NCUR
27
NCUR
28
 
29
 
30 31

1

2
3
4
5
 
6 7 8
Quiz 5
9

10
Univer. Day

11
 
12
 
13

14
15
16
17

18
19 20 21 22
 
23

24
Quiz 6
25
26 27 28

29
Review
30
1 Classes End
Review
2
3 4 5
6  
 
7  Final
6-7:50pm
8
9

 

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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.