Math 267 Elementary Differential Equations and Laplace Transforms Section B1

New Information

Course grades are now available. Solution to the final will be posted later this week.

Homework assignments (in reverse chronological order):
Date        Section       Problems (due next period unless stated otherwise)
5/3               5.2,3        review
5/2               5.2           1,2,3,5: for y(x0)=-2, y'(x0)=1, find a0, a1, ..., a5.
                                   Plot the solution using partial sums through a4x^4 and through a5x^5 over -1.5<x<1.5
4/30            5.2            5,6
4/29            5.2            1,2,3
4/28            5.1            19, 21
4/25            5.1            1,2,3,10,11,13,14
4/24            6.5, 6.6       6.5:1,2,3,13
4/22            6.4            5
4/22            6.4            1,2,3,14,15
4/19            6.3            1-7
4/15            6.3
4/12            6.2            11-13
4/11            6.2            1-5,7
4/9              6.1            1,2,3 5ab
4/8              7.9, 6.1
4/5               7.9            1,2 and initial value problem
4/4               7.9
4/2              7.7
4/1            7.4, 7.7        7.7: 1,3,5,11
3/29            7.8                2,3,5,7,8
3/28            7.8                7.6: 4(hand in Monday 4/1), 25, 26; 7.8: 1
3/26            7.6                 2,3,8 phase planes, 7,10
3/25            7.6                2,3,8,9
3/15           7.6                7.3: 18; 7.5:10; 7.6:1
3/14           7.5                due 3/25: 5,6,14,16,18,29
3/8            7.5                due 3/14: 1,11,12,13,15,17
3/7            7.3                15,19,22,24
3/5            7.3                1-7
3/4           7.2                4,6,10,12,13,14,18
3/1           7.2                1,2,3
2/28        3.2, 3.3            3.2: 7,8,14,16; 3.3:1,2,3
2/26 LECTURE NOTES (including homework) for Tuesday 2/26
2/25         3.7               5,6
2/22        3.6                5,6 (also review problems 3.1: 7,11,16; 3.4:11,16,24; 3.5:1,12,14)
2/21        3.6                1,2,4,13,17
2/19        3.8                1,2,3
2/18        3.8                5, 6, 9, 10, 11 + finish in-class example
2/15        3.8                8,12
2/14        3.4/5             3.4: 17,18,19; 3.5: 2,3,4,5,11,13
2/12        3.4                7,8,9,10,12
2/11        3.2                1,2 due 2/12, 7, 8 due later
2/8          3.1                1,3,5,6 due 2/11, 9,10,17,20 due 2/12
2/1          2.9                2,5,9
2/1          2.6                7,8,19,25,27
1/31        2.4                1, 2, 3
1/29        2.7                handout, 1, 5, 6, 13
1/28        2.5                3,4,6, 14, 15a, 20, 21 (due 1/31)
1/25        2.5                handout
1/24        2.3                1,3,9,14,20,21,24,29 (due 1/28)
1/22        2.1                14,16,19 (was collected 1/24)
              2.1                1,13
              2.6                1-3,7,9,11
              2.2                2,3,6,7,9,11
              1.3                1,5,7,8,9
              1.2               1,3,13
              1.1                1

Review for Test 1 Tuesday Feb. 5 over Chapters 1 and 2:
You may use a 1 page (1 side) help sheet (that you prepare) for the test. You may also use a table of integrals if desired, but you need to give it to me by Monday 2/4 with your name on it (it will be returned to you at the test). The test will cover basic setup or solution of 1st order differential equations. Explicit solutions are preferred to implicit, but explicit are not always possible. Basic material includes: (makes sure you can work problems of the following types)

1st order linear: explicit solution (general and IVP) such as 2.1 homework, e.g., 2.1.16 (1st order linear formula is important)
Applications of 1st order linear: mixture/pond (2.3.1), exponential growth/decay (2.3.14), motion (2.3.22)
1st order separable: implicit solution (general and IVP), such as 2.2 homework, e.g., 2.2.2, note separable includes autonomous
population models, stable equilibria (exponential growth, logistic equation, US population examples): handout homework. Logistic equation solution explicit formula is important.
exact equations: determine whether exact and if so find implicit solution (general and IVP), such as 2.6.7
use of integrating factors to convert nonexact to exact, e.g., 2.6.19, finding integrating factors of 1 variable, e.g., 2.6.25
Euler's method of numerical solution (requires calculator program, question will be optional)
difference equations: solution of 1st order linear, e.g., 2.9.2, and interest with payments e.g., 2.9.9

It is important you know how to go back and forth between the dy/dx = f(x,y) form of a differential equation and the M(x,y)dx + N(x,y)dy = 0 form.
ALL the differential equations on the test will appear in one form, which may NOT be the form in which homework problems from that section were presented. There may also be extra-credit questions covering additional material.

Review for Test 2 Tuesday March 12 over Chapter 3 and 7.2, 7.3:
This test requires a calulator capable of evaluating the reduced row echelon form (rref) of a matrix. You may use a 1 page (1 side) help sheet (that you prepare) for the test. You may also use a table of integrals if desired, but you need to give it to me by Monday 3/11 with your name on it (it will be returned to you at the test). The test will cover 2nd order linear differential equations and matrix algebra. Basic material includes: (makes sure you can work problems of the following types)

2nd order linear homogeneous with constant coefficients (general and IVP, all 3 types) such as 3.1:2,16; 3.4:11,20; 3.5:6,12
2nd order linear nonhomogeneous by undetermined coefficients (one specific solution, general solution, IVP) such as 3.6:2, 3, 3, 13, 19a
Applications of 2nd order linear homogeneous: springs and circuits. 3.8: 7, 9(omit last 3 sentences), 12(also plot)
Existence and Uniqueness of solutions to 2nd order linear differential equations, linear independence of a fundamental set of solutions & Wronskian 3.2:4, 8(explain), 14, 16; 3.3: 5, 11, 13, 24, 27
Matrix arithmetic (7.2:1,13)
Solution of systems of linear (algebraic) equations (7.3:1,4)

Eigenvalues and eigenvectors of matrices (7.3: 23, 24)

Solutions to Second Test

Review for Test 3 over Chapter 7 and 6.1, 6.2:
This test requires a calculator capable of evaluating the reduced row echelon form (rref) of a matrix. You may use a 1 page (1 side) help sheet (that you prepare) for the test. You may also use a table of integrals if desired, but you need to give it to me by Tuesday 4/16 with your name on it (it will be returned to you at the test). The test will cover systems of 1st order linear differential equations and derivation of Laplace transforms and computation of inverse Laplace transforms. Basic material includes: (makes sure you can work problems of the following types)

linear homogeneous with real eigenvalues and "enough" eigenvectors (general and IVP) such as (solution only) 7.5: 2-4,7,14-16
linear homogeneous with complex eigenvalues (general solution, IVP) such as (solution only) 7.6: 1, 2,3, 9, 10
linear homogeneous with repeated real eigenvalues (general and IVP) such as (solution only) 7.8 1,2,3,5, 8-10
linear independence and fundamental matrices such as 7.7 2, 4
nonhomogeneous systems 7.9
interpretation of phase planes
applications of linear homogeneous: circuits such as 7.2: 25, 26
derivation of the Laplace transform of a function such as 6.1: 5ab, 6
computation of inverse Laplace transform, such as 6.2: 3,4,8,9

Solutions to Test 3

Review for Chapter 6 and Sections 5.1, 5.2
Laplace transforms and series solutions to differential equations.
Basic material includes: (makes sure you can work problems of the following types)

Laplace transforms of functions and piecewise continuous functions such as 6.1:1, 6.3:6, 7
inverse Laplace transform such as 6.2:5,8, 6.3:7,
solution of a differential equation with initial conditions by Laplace transforms such as 6.2: 12,13, 6.4: 2,3, 6.5:2,3
series solution (general and IVP) of a differential equation at an ordinary point (partial sum through x^5 term and graph of solution of IVP) such as 5.2:5,15, 16
radius of convergence of a power series such as 5.1.4

Review Session Monday May 6, 9:45-11:45 AM in the regular classroom
Exam week office hours: (in addition to review session)
Wed. May 8, 9:30-11AM, Thurs. May 9, 9-11AM

The Final Exam was at the time for the Math 267 group (Thursday, May 9, 4:30-6:30 PM) in 2245 Coover.

Final exam
This test requires a calculator capable graphing and of evaluating the reduced row echelon form (rref) of a matrix. A table of Laplace transforms will be provided. You may use a 3 page (3 sides total) help sheet (that you prepare) for the test. You may also use a table of integrals if desired, but you need to give it to me by Wednesday 5/8 with your name on it (it will be returned to you at the test).
This exam will have 2 parts:
Part 1 will resemble previous tests and will have material on Ch 1-2 and Ch 5-6. (60-70% of the test)
Part 2 will have longer questions on material from Chapters 3 and 7. You will have a choice as to which questions in part 2 to answer. (30-40% of the test)