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     Math 284 Differential Equations (Spring 2021)

        Instructor  
  Mark Pernarowski 
        Textbook   Differential Equations (9th ed.), Nagle, Saff, Snider
        Office Hours   Schedule  (email to schedule an Online meeting)
        email   pernarow @ montana.edu
        Lectures  

MWF 11:00-11:50am  (Face-Face Lewis 304)
           

T        11:00-11:50am  (Online WeBeX)
           

T        is just another lecture but online.

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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 scores will be
posted in D2L.


Exam and quiz dates are indicated
on the schedule below. Their content
will be announced in class and
posted below

All exams/quizzes are closed book.

No electronic devices/phones
are permitted.
 

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.

Additional review handouts and lecture
notes will be posted in D2L under 
"Contents" as the course develops.
     

Covid Policy 

University Policy

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Schedule

Below is a calendar showing the schedule of quizzes (orange) and exams (red) and  holidays (green).
An approximate schedule of textbook section numbers is included and will be updated.

 

Sunday Monday Tuesday Wed Thursday Friday Saturday

10

 11       (1.1)
Classes start		 12        (1.2) 13       (1.2)/(2.1) 14

15      (2.2)

 16
17

 
18

MLK Day

 19        (2.2)/(2.3)

20        (2.3)/(2.4)

21

22       (2.4)
Quiz 1

23   
24
 

25       (2.6)

 26        (2.6) 27       (2.6) 28 29        (3.2) 30
 
31

1        (3.2)

 2          (3.4)
 3         (3.5) 4
5.       (4.1)
Quiz 2
 6
7
 

8       (4.2)

 9       (4.3)
10
Review		 11 12
Midterm 1		 13
14
 
15
Pres. Day
 16        (4.2) 17      (4.2)

 18 19       (4.4) 20
 
21
 
22       (4.4)
 23    (4.5)/(4.6) 24     (4.6)
 25 26       (4.7)
Quiz 3		 27
 
28 1         (4.7)
 2         (4.7)
 3      (4.9) 4 5  (4.9)/(4.10) 6
 
7
 		 8        (7.2)

 9        (7.2)
10      (7.3)
 11

12       (7.3)
Quiz 4

 13
 
14
 		 15     (7.4)

 16      (7.4)
17
Review		 18

 19
Midterm 2
 20
 
21
 		 22    (7.6) 23     (7.8)

 24  (7.8)

 25
26  (9.1-9.3)
 27
 
28
 		 29 (9.1-9.3) 30  (9.1-9.3)
 31  (9.4)
Quiz 5
 1
2
Univer. Day
 3
4 5 (9.4) 6          (9.5) 7         (9.5) 8

9       (9.6)

 10
 
11
 		 12     (9.8)

 13        (9.8)
14      (9.7)
 15
16       (9.7)
Quiz 6
 17
18 19     extra 20      extra 21      extra
 		 22

 23     extra

 24
25

26
Review

 27
Review
28
Final 11am
29
30
no class
 1

 

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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 POSTED IN D2L
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 IS NOW POSTED IN D2L
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
TBA
 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
TBA
 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.
 Linear Systems - Generalized eigenvectors and Repeated eigenvalues.
 

Final: Wed. April 28, 11:00-11:50am

           Lews Hall 304 (regular class)

Chapter  9 : description below

Review problems posted in D2L

 

 

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Exam and Quiz Content Descriptions:

 
   Quiz 1                            1.1,1.2,2.2

Basic definitions (linear, order,......), explicit and implicit solutions of differential equations, verifying y(x) is a solution, finding the 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) nor the Taylor series method in Chapter 1.
    Quiz 2   2.3,2.4,2.6

know separable, linear, exact, homogenous, Bernoulli definitions  and solution techniques. One question will be a chart where you decide if an exact is linear, homogenous, separable, Bernoulli. Three questions will be straight up solutions of first order ODE's of the aforementioned types.
    Quiz 3   4.2,4.3,4.4,4.5 Finding homogeneous solutions to all second order equations and possibly to a simple third order equation which is easily factorable. Finding particular solutions using Method of Undetermined coefficients. Lastly, finding general solutions (yh + yp) and solving IVPs.
    Quiz 4   4.6,4.7 Homogenous Cauchy-Euler equations and initial value problems. Variation of Parameters to find particular solutions given homogeneous solutions (you may need to compute the Wronskian). Reduction of Order to find second homogeneous solution and hence general homogeneous solution. NO questions on mechanical vibrations or Laplace transforms.
    Quiz 5  
7.2,7.3,7.4,7.5,7.8
 Laplace transform definition, finding transforms and inverse transforms, solving initial value problems, convolution theorem. There won't be anything from 7.6, 7.7, 7.9 of the text. You are allowed the Laplace Transform Summary sheet posted in D2L.
    Quiz 6   9.4,9.5,9.6,9.8

Fundamental matrices, general solns and solving Initial Value Problems. For the 2 by 2 matrix A, finding a general solution of x'=Ax where A has i) real distinct eigenvalues, complex eigenvalues and real repeated eigenvalues

 

 

 

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Midterm 1 

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 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 an application problem: Only a mixing problem  (15%)
  • There will be no questions on other applications: Newtonian Mechanics, Circuits,...
  • One question will require you to categorize types of differential equations (15%).
  • The sample problems posted in D2L are a good indication of the difficulty level of the problems.
  •  

 

 

 

 

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Midterm 2  Content Description

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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 posted in D2L are a good indication of the difficulty level of the problems but this  sheet has only one amplitude-phase problem.
  • Note: Undetermined coefficients is ONLY for L(y)=ay''+by'+cy=f and not L(y)=ax2y''+bxy'+cy=f
  • There will be no Laplace Transform questions

 

 

 

 

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Final: Content description

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Wednesday April 28 - 11:00-11:50am in Lewis Hall 304 (regular class location)

Material from sections 9.4-9.8 of the textbook

Topics Covered

  • 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: 01/05/2021.