Math 274 Differential Equations (Spring 2016)

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

 

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:

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.

Midterm 2

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 y_h(t)
2. Constant Coefficient 3rd order homogeneous y_h(t) with one solution known
3. Constant Coefficient 2nd order: Undetermined Coefficients Method for y_p(t)
4. General Solutions y(t)=y_h(t) + y_p(t), Initial Value Problems, Wronskian for independence
5. Cauchy Euler 2nd Order homogeneous y_h(t)
6. Variation of Parameter Method for y_p(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

Final

Important: Thursday, May 5, 2:00-3:50pm, Room ROBH101

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

5. 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.