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                           Math 284 Differential Equations (Spring 2018)

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    Math 284

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        Instructor  
  Mark Pernarowski 
        Textbook   Differential Equations (9th ed.), Nagle, Saff, Snider
        Section   01
        Office Hours   Schedule (Wil 2-236)
        Phone   994-5356
        Classroom  

MTWF 8:00-8:50am (Wil 1-144)

        URL   www.math.montana.edu/pernarow/M284

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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
7

8

9 10
Classes start
11 12 13
 
14
 
15
MLK Day
16 17

18
19
Quiz 1
20
 
21
 
22 23 24 25 26 27
 
28 29 30
31
1
2
Quiz 2
3
4
 
5

6

7
Review
8

9
Midterm 1
10
11
 
12
13 14

15 16 17
 
18
 
19
Pres. Day
20 21

22 23
Quiz 3
24
 
25 26
27
28 1 2 3
 
4
 
5

6
7
Quiz 4
8

9 10
 
11
 
12
Spring Break
13
Spring Break
14
Spring Break
15
Spring Break
16
Spring Break
17
 
18
 
19 20

21
Review
22

23
Midterm 2
24
 
25
 
26 27

28

29
30
University. Day
31
1
 
2 3 4
 
5

6
Quiz 5
7
 
8
 
9

10
11
12
13
LastDropDay
14
15 16 17 18
 
19

20
Quiz 6
21
22 23 24

25
Review
26
27 Classes End
Review
28
29 30 1
2   Final
    8-9:50am
3
4
5

 

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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 1,2,3,5,7,8,9,11,17,18,19,23  1rst Order Separable
2.3 2,3,4,7,9,10,13,15,17,18,19,22  1rst Order Linear
2.4 1,2, 5 (solve as well),11,12,13,22,25,26  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
     Review for Quiz 2 - Misc Problems
3.1 none  Mathematical Modelling
3.2 1,3,7  Mixing models (only)
3.3 1,3,5  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 Problems
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: 45,47  Cauchy-Euler equations, Reduction of Order
4.8 not covered  Qualitative theory
4.9 1,7,9,11  Mechanical Vibrations
4.10 Not on exam  Mechanical Vibrations: Forced
  Midterm 2  Chapter 4 on HW material assigned
     Review questions
5 Time permitting at end of course  Phase Plane, Numerical
6 not covered  General Theory of Linear Equations
  Laplace Transform Table Will be supplied by me at quizzes and final exam. DO NOT bring your own copy.
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 1,2,3,5,7,9,13  Laplace Transform Convolution Theorem
7.8 not covered  Laplace Transform - delta function
7.9 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  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 Class notes and review problems  Linear Systems - Repeated eigenvalues.
  Final  Chapter 7 and 9
     Review questions: Laplace Transforms  and Systems
     

Exam and Quiz Outlines (Tenative)

Quizzes (Tenative)

    Quiz 1                sample     1.1,1.2,2.2  
    Quiz 2 sample     2.3,2.4,2.6 (you will be asked to solve an exact, 
a linear, a homogenous and a Bernoulli eqn)
    Quiz 3 sample     4.2-4.3, 4.4 Will include general soln of higher order constant coefficient eqns and simple problems on undetermined coefficients (4.4). Will NOT include theory regarding independence, Wronskians, etc.
    Quiz 4 sample     4.6,4.7 and Reduction of Order Cauchy-Euler homogeneous, Variation
of parameters, Reduction of Order
    Quiz 5 sample     7.2,7.3,7.4,7.5 (not 7.7) Bring your Laplace transform table!! You will have to take transforms using tables and transform properties, inverst transforms and solve IVP. Partial fractions will be at most "cubic".
    Quiz 6 sample     Convolutions 7.7,9.4,9.5

DO NOT BRING YOUR LAPLACE TABLE TO THE QUIZ.

I WILL DISTRIBUTE ONE WITH THE QUIZ AND COLLECT IT FROM YOU AT THE END

YOU CAN NOT USE ANY ELECTRONIC DEVICE. THAT INCLUDES CELLPHONES, IPODS, IPADS, CALCULATORS, ETC.

IF I SEE YOU WITH ONE I WILL TAKE YOUR TEST AWAY AND YOU WILL GET A ZERO

The focus on 9.4-9.5 will be: can you solve IVPs and can you find general

solutions for the distinct eigenvalue case?

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Midterm 1 (Tenative)

Sample Problems

 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 are a good indication of the difficulty level of the problems.

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Midterm 2 (Tenative)

Sample Problems

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
  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 unforced 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 problems 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

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Final (Tenative)

Tuesday, May 2, 8:00-9:50am, Wil 1-144

  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: 08/07/2015.