Math 274 Differential Equations (Spring 2016)
Final Grades have been submitted and available through MyInfo (not D2L)
If you have questions email me: pernarow@math.montana.edu
Have a great summer!
Instructor |
Mark Pernarowski | |||||
Textbook | Differential Equations (8th ed.), Nagle, Saff, Snider | |||||
Section | 09 | |||||
Office Hours | Schedule (Wil 2-236) | |||||
Phone | 994-5356 | |||||
Classroom |
MF 1:10-2:00pm (Herrick Hall HH 313) TR 12:40-1:30pm (JONH 213) |
|||||
URL | www.math.montana.edu/pernarow/M274 |
|
Grading: The course % is determined by: |
Syllabus: Material covered in text is from:
|
Schedule
Below is a calendar showing the schedule of quizzes and tests (red) and holidays
(green).
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
Quizzes
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? |
Midterm 1
Sample Problems
The exam will cover material from the following sections of the textbook:
- Section 1.1 ODE definitions and theory
- Section 1.2 IVP explicit/implicit solutions, existence uniqueness
- Section 2.2 Separable Equations
- Section 2.3 Linear Equations
- Section 2.4 Exact Equations
- Section 2.6 Homogeneous Equations and Bernoulli Equations only
- Section 3.2 Mixing Problems (no population problems)
- 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.
Midterm 2
Sample Problems
The exam will cover material from the following sections of the textbook: 4.2-4.7, 4.9
- Constant Coefficient 2nd order homogeneous yh(t)
- Constant Coefficient 3rd order homogeneous yh(t) with one solution known
- Constant Coefficient 2nd order: Undetermined Coefficients Method for yp(t)
- General Solutions y(t)=yh(t) + yp(t), Initial Value Problems, Wronskian for independence
- Cauchy Euler 2nd Order homogeneous yh(t)
- Variation of Parameter Method for yp(t) - standard form.
- Reduction of order: homogeneous solution y2(t) from given homogeneous y1(t)
- 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
Final
Important: Thursday, May 5, 2:00-3:50pm, Room ROBH101
- Sections of the textbook covered: 7.2-7.5, 7.7, 9.4-9.7 and repeated eigenvalues notes
- Sample Problems: Laplace Transforms (Ch 7) and Systems (Ch 9)
- Laplace Transform Table is attached to test.
DO NOT BRING YOUR OWN!! - The exam will be about 60% on systems and about 40% on Laplace transforms
-
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
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.