Math 225 - Introduction to Differential Equations

Class: MW 2:30PM - 3:45PM in Math & Psychology 106 (8/27-12/9/2025), Instructor: Bedrich Sousedik

Topics of this course include: solutions of first- and second order linear differential equations, non-linear exact and separable equations, integrating factors, homogeneous equations, higher-order linear equations, initial and boundary value problems, solutions as functions of the equation parameters, Laplace transforms, power series solutions for Bessel and Legendre equations, difference equations and numerical methods.

Recommended Preparation: MATH 251.

The detailed class syllabus (pdf) is here. However, look below to see the actual progress of the class.

errata

8/27 Introduction, (1.1-1.2) Classification of differential equations, ODEs.

9/1 Labor Day (no class).

9/3 (1.2) IVP, Picard's theorem, (2.1) First-order linear equations.

9/8 (2.1) First-order linear equations. Hw1 (due 9/15): 1.1/6,8; 1.2/7,11,15,41,42; 2.1/12,14,20.

9/10 (2.2) Separable equations.

9/15 Some applications: (2.3) Growth and decay. Quiz 1. Hw2 (due 9/22): 2.1/3,18,19; 2.2/22,24,29,30.

9/17 Some applications: (2.5) cooling and heating phenomena.

9/22 More applications of the first-order equations: continuos compounding. Hw3 (done in class, no quiz): 2.3/11,17,36 and 2.5/9,11.

9/24 (3.1) Introduction to second-order linear equations, (3.2) Fundamental solutions of the homogeneous equation. Hw4 (due 10/6): 3.1/17,18,43 and 3.2/5,6,20.

9/29 (3.3) Reduction of order. (3.4) Homogeneous equations with constant coefficients: real roots.

10/1 (3.4) Homogeneous equations with constant coefficients: real roots. (3.5) complex roots. Hw5 (due 10/13, Exam #1): 3.3/7,8,11, 3.4/14,18, 3.5/10,12, 3.6/8,10,12.

10/6 (3.6) Nonhomogeneous equations. (3.7) Solving nonhomogeneous equations: method of undetermined coefficients.

10/8 (3.7) Solving nonhomogeneous equations: method of undetermined coefficients.

10/13 Exam #1 (1.1-3.6).

10/15 (3.7) Solving nonhomogeneous equations: method of undetermined coefficients. Hw 6 (due 10/22): 3.7/7,9,11,14,17.

10/20 (3.8) Solving nonhomogeneous equations: method of variation of parameters. Hw 7 (due 10/27): 3.8/2,8,9,16.

10/22 (3.9) Mechanical systems. Simple harmonic motion.

10/27 (3.10)-(3.11) Unforced and forced vibrations.

10/29 (5.1) Definition of the Laplace transform. (5.2) Properties of the Laplace transform: (i) derivative property.

11/3 (5.2) Properties of the Laplace transform: (ii) translation property, (iii) derivative of transform. (5.3) The inverse Laplace transform. Hw 8 (due 11/10, Exam #2): 5.1/20,24,36 and 5.2/3,5,12,17,38,41,42.

11/5 (5.3) The inverse Laplace transform. (5.4) Initial-value problems using Laplace transform.

11/10 Exam #2 (Sections 3.7-5.2 as covered in class and hw, no applications or word problems).

Plan:

11/12 (5.4) Initial-value problems using Laplace transform. (5.5) Step functions and delayed functions: introduction. Hw 9 (due 11/19): 5.3/13,19,20 and 5.4/8,12,14.

11/17 (5.5) Step functions and delayed functions: Laplace transform of a delayed function. (5.6) DE with with discontinuous forcing functions.

11/19 (5.6) DE with with discontinuous forcing functions. Hw 10 (due 12/1): 5.5/4,8,14,21 and 5.6/4,11,12.

11/24 and 11/26 Thanksgiving (no class).

12/1 (5.7) Impulse forcing functions. Hw 11 (due 12/8) : 5.7/6,11,13.

12/3 (5.8) The convolution integral.

12/8 (5.8) The convolution integral: transfer function. Modeling (and word) problems.

Friday 12/12, 1-3pm in MP 106 Final (comprehensive).