Final: Thursday, May 5, 2:003:50pm, Room ROBH101
Instructor 
Mark Pernarowski  
Textbook  Differential Equations (8th ed.), Nagle, Saff, Snider  
Section  09  
Office Hours  Schedule (Wil 2236)  
Phone  9945356  
Classroom 
MF 1:102:00pm (Herrick Hall HH 313) TR 12:401:30pm (JONH 213) 

URL  www.math.montana.edu/pernarow/M274 

Grading: The course % is determined by: 
Syllabus: Material covered in text is from:

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  CauchyEuler 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(t2)),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  4.24.3, 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  CauchyEuler 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 9.49.5 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: 4.24.7, 4.9
 Constant Coefficient 2nd order homogeneous y_{h}(t)
 Constant Coefficient 3rd order homogeneous y_{h}(t) with one solution known
 Constant Coefficient 2nd order: Undetermined Coefficients Method for y_{p}(t)
 General Solutions y(t)=y_{h}(t) + y_{p}(t), Initial Value Problems, Wronskian for independence
 Cauchy Euler 2nd Order homogeneous y_{h}(t)
 Variation of Parameter Method for y_{p}(t)  standard form.
 Reduction of order: homogeneous solution y_{2}(t) from given homogeneous y_{1}(t)
 Mechanical Vibrations: Amplitude Phase Form y= A sin(wt+phi) for unforced case
Notes:
 There will be an amplitudephase problem (1015%). In fact, there will be a question from each point 18 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 amplitudephase problem.
 Undetermined coefficients is ONLY for L(y)=ay''+by'+cy=f and not L(y)=ax^{2}y''+bxy'+cy=f
Final
Important: Thursday, May 5, 2:003:50pm, Room ROBH101
 Sections of the textbook covered: 7.27.5, 7.7, 9.49.7 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.