For the most recent information go to the end of the page.

Remember that the final exam is on May 4, 14:00, SE Cafeteria.

Office hours are the day before the exam, on May 3, 14:00-16:00.

Solutions for all quizzes are posted.

Office hours for study week are arranged.

Meera Gupta(rm 3093F)

April 16	® 10 to 12
April 26	® 1 to 3
May 3	® 12 to 2

Alex Shlakhter (rm 3093H)

April 26	® 10 to 12
May 3	® 10 to 11

Also, after April 8th all the leftover tests quizzes will be in a box in Math help centre (3093C)

Notice that you probably need to refresh this page to get the most recent information.

INSTRUCTOR: Dmitry Khmelev

OFFICE: SB 4059B dkhmelev (at) math.toronto.edu

OFFICE HOURS: Tuesday 12:00-13:00, Thursday 12:00-13:00 or by appointment.

REQUIRED TEXTBOOK: Elementary Differential Equations, 4th edition, H.Edwards and D.Penney.

EVALUATION: There would be one Midterm test worth 20%. There will be quizzes (tentatively each second week) worth 30%, and a comprehensive final exam worth 50%.

TIME&PLACE: Tue 11:00-12:00 & Thu 10:00-12:00, SE 2072.

TUTORIALS: Your TAs are Meera Gupta (mgupta@utm.utoronto.ca, SE 3093F) and Alex Shlakhter. The tutorials times and places are:

MARKS: Marks will be posted behind the glass of 3093F. All the latest marks are to be seen there.

MISSED QUIZ: There is no make-up quiz. If you miss a quiz due to illness, you will have to bring a dated doctor's certificate stating that you were too ill to take the quizzes. In that event we will re-rate your score based on the score you take. If, for example, you miss two quizzes (out of say six), and have a valid medical reason for both of missed quizzes, then your score will be computed based on the average on the four quizzes you took.

MID-TERM TEST: On Mar 2nd, 2004 (in-class for 50 minutes).

MISSED TEST: There is no make-up test. Students who cannot be present for a test must contact their instructor in person on or before their first day back at school. They must also bring a dated doctor's certificate stating that they were too ill to attend the school on the day of the test and giving the expected duration of their illness. Besides illness only very serious reason, properly documented, can be considered as valid excuses for missing a test. In these cases the missed test mark will be calculated by taking 40% of the final exam mark. All other cases will receive a mark of zero on the missed test. There will be no supplemental exam.

DROP DATE: March 7, 2004

RE-MARKING THE TEST:

• Any oversight in the marking of a test paper should be brought to the attention of your tutor immediately after the test was returned.
• All matters regarding test marks (e.g. additional errors) must be communicated to me, outside the class, within one week of return date.
• No test written in pencil or erasable pen may be submitted for re-marking.

CHANGES: Changes to and clarification of details in this course outline will be announced to the class on at least two occasions to take effect.

COURSE OBJECTIVE: This course presents the basic ODE methodology used in many fields of applications. The emphasis of this course is on concepts and techniques and will be useful to students who seek to gain an understanding of the use of ODE in their own field.

SYLLABUS: 1.1, 1.2, 1.3 (Existence and Uniqueness), 1.4, 1.5, 1.6; 2.1, 2.2, 2.3, 2.5, 2.6, 2.7; 5.1, 5.2, 5.3, 5.4, 5.5. If time permits we will do selected sections from Chapters 3 and 4.

Recommended problems for quiz #1, Thursday, Jan 15, 2004. The quiz will hold in the beginning of the lecture and will take 25 minutes:

Section 1.1. Verification by substitution (1-16); determining constants from initial condition (17-26); text problems (32-36); solving ODE by educated guess (37-42); investigation (43).

Section 1.2. Simple integration (1-18).

Section 1.3. Existence and Uniqueness (21-31).

Section 1.4. General solution by separating variables (1-18), initial value problems by separating variables (19-28), exponential ODE and half-life (29-42), derivation of ODE (47).

Quiz #1 with solution: q/1.ps, q/1.pdf.

Recommended problems for quiz #2, Thursday, Jan 29, 2004. The quiz will hold in the beginning of the lecture and will take 25 minutes:

Section 1.5. Integrating Linear 1st order ODE (1-25); various problems (26-32), solution problems (33-37).

Section 1.6. Integrating using substitution methods (1-30); identifying and integrating exact equations (31-42).

Quiz #2: q/2.ps, q/2.pdf. Solution: q/2sol.ps, q/2sol.pdf

Recommended problems for quiz #3, Thursday, Feb 12, 2004. The quiz will hold in the beginning of the lecture and will take 25 minutes:

Section 2.1. Superposition principle and finding particular solution from general for linear ODE of 2nd order (1-16), superposition fails for non-linear ODE (17-19), linear independence (20-26), non-homogeneous ODE, general sol'ns, Wronskian and Ex&Uniq Thm (27-31), general solutions for homogeneous ODE (33-42), detecting 2nd order homogeneous ODE by its general solution (43-48).

Also, please find general solution for the following ODE's:

A1. y¢¢+16y=0.¹

A2. y¢¢-2y¢+5y=0.²

A3. 4y¢¢+8y¢+5y=0.³

In the following problems, find a homogeneous ODE of 2nd order with constant real coefficients, which has the given particular solution, or explain why such an ODE does not exists.

A4. pcos(3x)e^-2x.⁴

A5. (sin(2x)+1)e^-2x.⁵

A6. (sin(2x)+cos(2x))e^-2x. ⁶

A7. xe^-2x. ⁷

A8. e^-2x+x.⁸

Section 2.2. Linear dependence by linear combination (1-6), linear independence via Wronskian (7-12), initial value problem from three linearly independent solutions (13-20), various problems (21-34).

Quiz #3: q/3.ps, q/3.pdf. Solution: q/3sol.ps, q/3sol.pdf

Information for Midterm Test, Tuesday, Mar 2nd, 2004. The test will hold in-class, strictly for 50 minutes. No aids allowed. The test will be similar to quizes, with 6-7 questions. The questions will be on Sections 1.1-1.6; 2.1-2.3. Additionally to exercises on previous sections you should train on the following problems:

Section 2.3: General solutions for constant-coefficients ODE (1-20, 27-32), initial value problems (21-26,37,38), identifying a root by particular solution (33-36), idenifying ODE by general solution (39-42).

Additional office hours before midterm: Mon, Mar 1st, 11-13.

Solution manual: http://www.prenhall.com/divisions/esm/app/edwards/ssm/edeq.html

Midterm: q/Midterm.ps, q/Midterm.pdf.

Additional office hours before midterm: Mon, Mar 1st, 11-13.

Solution manual: http://www.prenhall.com/divisions/esm/app/edwards/ssm/edeq.html

Recommended problems for quiz #4, Thursday, Mar 11th, 2004. The quiz will hold in the beginning of the lecture and will take 25 minutes:

Section 2.5, Nonhomogeneous ODE's: finding particular solution by variation of parameters (1-20), identifying form of particular solution (21-30), solving initial value problems (31-40)

Section 2.6: Representing particular solution as a sum of two oscillations (1-6), finding steady periodic solution (7-14), identifying practical resonance frequency in mechanical systems (15-18)

Section 2.7, electrical circuits: finding steady periodic current (11-16), finding formula for current I(t) (17-22)

Quiz #4: q/4.ps, q/4.pdf. Solution: q/4sol.ps, q/4sol.pdf

Recommended problems for quiz #5, Thursday, Mar 25th, 2004. The quiz will hold in the beginning of the lecture and will take 25 minutes:

Section 5.1: reduction to first-order system (1-10).

Section 5.2: elimination method for solving first-order systems (1-19).

Section 5.3: reviewing matrix arithmetics (1-8), representing first-order systems in matrix form (11-20), verification of particular solutions, and constucting the general solution, solving initial value problem (31-40), various problems on linear dependence (41-45).

Section 5.4: finding general solutions using eigenvalue method for 2×2 matrices (1-16), 3×3 matrices (17-26).

Quiz #5: q/5.ps, q/5.pdf. Solution: q/5sol.ps, q/5sol.pdf

Recommended problems for quiz #6, Thursday, Apr 8th, 2004. The quiz will hold in the beginning of the lecture and will take 25 minutes:

Section 4.1: computing Laplace transform by integration (1-6), by using table of precomputed transforms (13, 15-22); basic inversion (23, 26-32); miscellaneous problems (36).

Section 4.2: application of Laplace transform for solving initial value problem (1-16); inverting Laplace transform by integration formula (11-19); using Laplace transform of derivative for computing Laplace transforms of other functions (19-25).

Section 4.3: Translation theorem (1-10); Partial fractions inversion (11-22); Solving initial value problems (27-38).

Quiz #6: q/6.ps, q/6.pdf. Solution: q/6sol.ps, q/6sol.pdf

Recommended problems for exam preparation (in addition to the above questions).

Section 3.1: 1-6, 11-14.

Section 7.1: 1-6, 9, 10, 13, 21.

Recent results:

Last 5 dig of SN	Q/9	Q/10	Q/10	T/70	Q/8	Q/10	Q/10
00612	5.5	9	9	44.5	2	6.5	8
01745	8	8.5	8	62.5	6.5	9	8
02379
03091	3.5
03365	7.5		3	11
03819	6.5	9	8.5	58	8	10	10
05741	6	4		15		1
07017	5.5	5	3.5	19	2.5	1	6
07792	6	2	9	13
09142	6		2.5	13
09285
10510	8	4	8.5	38	2		10
10626	8.5	4	7	43	7	4	7
10901	0
11429
11450	6	9		42	8	3	8.5
11901	9	10	9	60	8	10	10
14321	7	5	8.5	45	0.5	2.5	1
21545	8	5	8.5	66	7.5	7.5	8
23717	6	2	8.5	29	3	4	0
23737	2	1	3.5
24595			10	35	2.5	2	4
24701	3	5.5	9	51	n	6.5
24737
25111	9	10	9	60	6	4.5	10
25120
25343	5.5	0	6	20	0	5.5	8
26040	6.5	9	10	44	8	9.5	8
26193	0	6	9	38	2.5	2	5
27530	6	9.5	6.5	66.5	2	5.5	7
27701	8.5	9	3.5	59	4	5	10
28085	5.5	6	4	46	5	7.5	8
29153	8.5	10	8	67	8	9.5	10
30188	5	5	8.5	19.5	0.5	2	3
32485
33950	3.5	5	8.5
34017	5.5	2	7.5	43	0.5		7
34261	n	9	8.5	39	n	3	9
34739	5.5	5.5	6.5	45	7	1	8
36335	7.5		4.5	42		4
36711	7.5	6.5	8.5	48	0	3
38360	6.5	3		47	2.5	6	10
41348	6	3	9	48	2.5	5
41383	9	9	8.5	70	7	9.5	10
41579	9	10	8.5	70	7.5	10	8
44277	5.5	10	8.5	59	6	8	10
45787		9.5	8.5	67	4	6	10
46615	0.5	6	8.5	55	4	6.5	8
46723	6.5	2.5	8.5	46	3	3.5	8
46804	6	5.5	9	48	4	3.5	8
47567	6	6	8	44	2.5	1.5	2
47634	5.5	5	5.5	22	2	1	6
47918	6.5	6.5	8.5	43	6.5	2	8
48115	8	7	9	50	6	5	8
48601	9	10	9	61	6	10	10
48970	2		5	25
48973	5.5	1.5	8	54	2.5	1	8
49348		2	5	31	0	2.5	7
49380	9	5	9	55	4	6.5	9
49507	8.5	10	9	68	6.5	7.5	8
49750				48
49828	6	6	9	46	6	5	9.5
50968	n	5	9	n	7	1
54366	9	10	9	65	7	9	10
55051	n	7.5	6	49	6
55409	6	8	6	35	4	1
56872		10	9	69	7	8	10
57025	9	5	6.5	n	6	4	10
58077
59158	5		9	31	2.5	3.5	8
59218	8		3	44	2.5	3	7
60770	6.5	6	4
60891	2.5	9	9	58	7	6.5	9
64169	9	10	8.5	67	8	5.5	5
64257	8	6	9	58	5	7.5	8
64381	6	7	9	63	2.5	2.5	10
65875	6	6	8.5	37	2	5	10
66188	5	10	8.5
67826	9	8	8.5	31	2	6	9.5
68272	8.5	5	n	33	n		0
68343	8.5	10	9	58
68832		3.5
68878	7.5	8		41	6	5.5	10
69414	5	6.5	3.5	57	4	5.5	8
70413	5.5	6	9	51	3		3
70484	n	8.5	10	52	7.5	3	n
71668	8	2	8.5	44
72233		8.5	8	56	4	4	5
72944	n			n
73391	7	6	8
73843		1	8	39	1	3	10
74596	9	n	9	n	6	5	n
75469	8	2	9	n	6	2.5	10
75487	6	9	8.5	58	5	9	10
76253	6	6	6	60	n	2.5	10
77158	9	6	8.5	40	4	8	8
78190	6	3.5	9	39	1.5	n	9
78342	6
79771	6	5		28	7	2.5	7
80178	5.5	2	9	34	0	5.5	10
81033	7.5	5	6	55	0	4	6
81623
83272	n	10	9	56.5	3	8	10
83980	3.5	2.5	9	56	3	6.5	10
84120
84593	6	6	9	64	6	6.5	9
85576	0	5	9	38.5	2.5	3	0
87474	8.5	7	8.5	32	7	3	7
88040	0		7	44	8	8	8
88296	9	10	9	69	8	10	10
89012	5.5	4	6	32	2	3.5	8
90837	5.5	4.5	5	60	0	1.5	10
91097	8	5	8.5	35	0	1.5
91120
91703
92085	8.5	7	9	55	1	6	10
93690	6.5	4	8.5	30	0
94314	5	5	6	52.5	4	4.5	1
94582	8	6.5	8	40	7	1.5	8
94804	6		6	40		2.5
94806	6	9		64	4	4.5	10
95304	6	3	3.5	21
95850	8	3.5	9	63	n	7	10
96162	8	9.5	9	69	3	10	9.5
96811	6	10	9	40	4	2.5	8
99066	6.5	5	5	48	3	3	6
99360	2	3	8.5		5	2	9
99437	2.5			19
99606	6	8	9	60	3	4.5	10
99964	2	5	8.5	67.5	2.5	6	9.5
99980	0	8.5	5	46	3	2	9

Footnotes:

¹C₁cos4x+C₂sin4x.

²(C₁cos2x+C₂sin2x)e^x

³(C₁cos(x/2)+C₂sin(x/2))e^-x

⁴From the form of the term we can read out the complex root for characteristic eqn -2+3i, which must appear together with the root -2-3i. Characteristic equation: (r-(2+3i))(r-(2-3i))=0 or r²-4r+13=0. Hence ODE is y¢¢-4y¢+13=0.

⁵First, group expression with respect to exponent: (sin(2x)+1)e^-2x=sin(2x)e^-2x+e^-2x. Term C₁sin(2x)e^-2x can only appear in a pair with term C₂cos(2x)e^-2x, since complex roots -2+2i and -2-2i come in a pair. To have the term C₃e^-2x we must also have root -2. Therefore the characteristic equation must have 3 roots -2±2i and -2, which is impossible for an equation of 2nd degree. Therefore there are no linear homogeneous ODE, having this particular solution.

⁶This particular solution fits general solution (C₁sin(2x)+C₂cos(2x))e^-2x with C₁=C₂=1. Such general solution is introduced by complex roots -2±2.

⁷This particular solution fits general solution (C₁+C₂x)e^-2x with C₁=0, C₂=1. Such general solution is generated by repeated root -2.

⁸Term e^-2x appears from the root -2. Term x appears in the expression (C₁+C₂x)e^0x, corresponding to the double root 0. Characteristic equation with 3 roots must be cubic, while for homogeneous ODE of 2nd order the characteristic equation is quadratic. Therefore such an ODE does not exists.