# Ordinary Differential Equations

CU Boulder

Instructor: Yuhao Hu

Email: yuhao.hu@colorado.edu

Office: Math 225

Office Hours: WF 4:00-5:45pm

Lectures: MWF 12:00-12:50pm at KCEN N252

## Overview

The subject of this course is ordinary differential equations (ODEs). These are equations that involve the rate of change of certain unknown function(s) with respect to a single independent variable, which is often time. Though there is a theoretical side of ODEs, this course is rather concerned with solution techniques and applications; hence, our goal is quite straightforward: to be able to identify the type of an ODE/system and solve it using an appropriate method.

## Textbook

*Ordinary Differential Equations*by M. Tenenbaum and H. Pollard

## Schedule

(weekly, only the first meeting of the week is dated; "Ls." stands for "Lesson" in the textbook.)**01/13**ODEs and their solutions (Ls. 1, 3); general/particular solutions (Ls. 4)

**01/22**1st-order ODEs, Direction field, phase line (Ls. 5A, additional reading); separable equations (Ls. 6C)

**01/27**1st-order ODE with homogeneous coefficients (Ls. 7); exact equations (Ls. 9, 10A);

integrating factors (Ls. 10B)

**02/03**Linear 1st-order ODE, integrating factors (Ls. 11A-C);

applications of 1st-order ODEs (a selection from Ls. 13, 14);

higher-order ODEs and complex numbers (Ls. 18)

**02/10**2nd-order Constant-coefficient Linear ODE, method of characteristic equations (Ls. 20);

higher order constant-coefficient Linear ODE (Ls. 20);

Constant-coefficient Linear ODE, non-homogeneous case (Ls. 21)

**02/17 (Midterm I on Wednesday)**Reduction of order (Ls. 23)

**02/24**Variation of parameters (Ls. 22); differential operators (Ls. 24)

**03/02**Laplace transform (Ls. 27A); Laplace transform, properties (Ls. 27B,D); the Laplace method (Ls. 27C)

**03/09**Step functions and delta functions (additional notes);

ODEs with discontinuous forcing terms (additional notes); Undamped Motion (Ls. 28)

**03/16**Damped Motion (Ls. 29); other 2nd-order problems (selection from Ls. 30M)

**03/23-27 (Spring break; no class)**

**03/30**

**04/06 (Midterm II on Wednesday)**

**04/13**

**04/20**

**04/27**

## Homework

(Due on each**Friday**, unless specified otherwise.)

**Homework 1**(Lectures 1-2) due Jan. 24

[Lecture 1]

**Reading:**pp. 1-4, pp. 20-23

**Exercises:**[p. 27,

**Ex3**: 1, 2, 3] (meaning: page 27, Exercise 3, questions 1, 2 and 3.)

[Lecture 2]

**Reading:**pp. 24-27, pp. 28-36

**Exercises:**[p. 27,

**Ex3**: 4] and [p. 37,

**Ex4**: 4, 6, 12, 14, 20, 28]

**Homework 2**(Lectures 3-4) due Jan. 31

[Lecture 3]

**Reading:**pp. 38-41, reading material 1 (link)

**Exercises:**[p. 45,

**Ex5**: 3, 5]

[Lecture 4]

**Reading:**pp. 52-55

**Exercises:**[pp. 55-56,

**Ex6**: 3, 4, 6, 7, 15, 18, 20]

**Homework 3**(Lectures 5-7) due Feb. 7

[Lecture 5]

**Reading:**pp. 57-60

**Exercises:**[p. 61,

**Ex7**: 1, 5, 6, 8, 10, 13]

[Lecture 6]

**Reading:**pp. 70-79

**Exercises:**[p. 79,

**Ex9**: 4, 7, 8, 13, 15]

[Lecture 7]

**Reading:**pp. 83-87

**Exercises:**[pp. 90-91,

**Ex10**: 3, 7, 8, 10, 12]

**Homework 4**(Lectures 8-10) due Feb. 14

[Lecture 8]

**Reading:**pp. 92-95

**Exercises:**[p. 97,

**Ex11**: 5, 6, 11, 14, 15, 17]

[Lecture 9]

**Reading:**pp. 107-110, pp. 117-118

**Exercises:**[p. 112,

**Ex13**: 2, 14] and [p. 120,

**Ex 14**11, 14, 15]

[Lecture 10]

**Reading:**pp. 196-203

**Exercises:**[p. 204,

**Ex18**: 4(b,g), 5, 6, 8]

**Homework 5**(Lectures 11-13) due Feb. 21

[Lecture 11]

**Reading:**pp. 211-220

**Exercises:**[p. 220,

**Ex20**: 1, 2, 14, 26, 33]

[Lecture 12]

**Reading:**no new readings

**Exercises:**[p. 220,

**Ex20**: 9, 15, 35]

[Lecture 13]

**Reading:**pp. 221-230

**Exercises:**[pp. 231-232,

**Ex21**: 3, 4, 6, 8, 10]

**Homework 6**(Lectures 14-15) due Feb. 28

[Lecture 14]

**Reading:**no new readings

**Exercises:**[p. 232,

**Ex21**: 28, 30, 31]

[Lecture 15]

**Reading:**pp. 241-246 (Note: The book uses different notations than we did in class.)

**Exercises:**[p. 246,

**Ex23**: 1, 3, 11, 12, 16]

**Homework 7**(Lectures 16-18) due Mar. 06

[Lecture 16]

**Reading:**pp. 233-240

**Exercises:**[p. 240,

**Ex22**: 7, 9, 12, 19]

[Lecture 17]

**Reading:**no new readings

**Exercises:**[p. 240,

**Ex22**: 14, 18]

[Lecture 18]

**Reading:**pp. 251-262

**Exercises:**[p. 266,

**Ex24**: 5(a,c), 6(c), 9(a), 10(a,c), 12(a), 13(a), 14(a,c), 15(a)]

**Homework 8**(Lectures 19-21) due Mar. 13

[Lecture 19]

**Reading:**pp. 263-265 and pp. 292-295

**Exercises:**[p. 267,

**Ex24**: 19, 33] and [p.311,

**Ex27**: 1, 2]

[Lecture 20]

**Reading:**pp. 295-303

**Exercises:**[p. 311,

**Ex27**: 13, 14, 15, 16, 18, 19]

[Lecture 21]

**Reading:**Laplace Transform I

**Exercises:**no new exercises

## Exams

**Midterm I:**2/19 Wednesday

Topics covered: Lectures 1-12

(A formula sheet allowed, A4 size, front and back, created on your own.)

Sample Exam // Solutions

Exam // Solutions

**Midterm II:**4/8 Wednesday

**Final:**TBD

## Grading

Two lowest homework grades will be dropped.

**Homework:**30%**Midterm I:**20%**Midterm II:**20%**Final Exam:**30%