## Course Overview

The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on linear differential equations and their applications in science and engineering. More details are given in the course goals below.

## Prerequisites

At MIT, 18.03 Differential Equations has 18.01 Single Variable Calculus as a prerequisite. 18.02 Multivariable Calculus is a corequisite, meaning students can take 18.02 and 18.03 simultaneously. From 18.02 we will expect knowledge of vectors, the arithmetic of matrices, and some simple properties of vector valued functions.

OCW users will find 18.01SC Single Variable Calculus and 18.02SC Multivariable Calculus useful in preparing for this course.

## Course Goals

By the end of the course students will be able to:

- Model a simple physical system to obtain a first order differential equation.
- Test the plausibility of a solution to a differential equation (DE) which models a physical situation by using reality-check methods such as physical reasoning, looking at the graph of the solution, testing extreme cases, and checking units.
- Visualize solutions using direction fields and approximate them using Euler's method.
- Find and classify the critical points of a first order autonomous equation and use them to describe the qualitative behavior and, in particular, the stability of the solutions.
- Use known DE types to model and understand situations involving exponential growth or decay and second order physical systems such as driven spring-mass systems or LRC circuits.
- Solve the main equations with various input functions including zero, constants, exponentials, sinusoids, step functions, impulses, and superpositions of these functions.
- Understand and use fluently the following features of the linear system response: solution, stability, transient, steady-state, amplitude response, phase response, amplitude-phase form, weight and transfer functions, pole diagrams, resonance and practical resonance, fundamental matrix.
- Use the following techniques to solve the differential equations described above: characteristic equation, exponential response formula, Laplace transform, convolution integrals, Fourier series, complex arithmetic, variation of parameters, elimination and anti-elimination, matrix eigenvalue method.
- Understand the basic notions of linearity, superposition, and existence and uniqueness of solutions to DE's, and use these concepts in solving linear DE's.
- Draw and interpret the phase portrait for autonomous 2x2 linear constant coefficient systems.
- Linearize an autonomous non-linear 2x2 system around its critical points and use this to sketch its phase portrait and, in particular, the stability behavior of the system.

The main equations studied in the course are driven first and second order constant coefficient linear ordinary differential equations and 2x2 systems. For these equations students will be able to:

## Format

The course, designed for independent study, has been organized to follow the sequence of topics covered in an MIT course on Differential Equations. There are four major units.

- First Order Differential Equations
- Second Order Constant Coefficient Linear Equations
- Fourier Series and Laplace Transform
- First Order Systems

Each unit is divided into sessions, which consist of written notes, lecture videos, problem solving videos, practice problems, and problem sets. Following the practice at MIT, the problem sets are split into two parts: Part I covering simple problems designed to emphasize a specific skill or technique, and Part II covering harder, often multistep problems, designed to help the student learn to apply the skills and techniques to more realistic problems. Complete solutions are provided for all problem sets.

To help guide your learning, you will see how problem solving is taught by an experienced MIT Recitation Instructor. At the end of each unit is an exam covering the material in the unit and a practice exam to help you prepare for the exam. Solutions are included for both the exam and practice exam.

At the end of Unit IV is a final exam covering the entire course.

MIT expects its students to spend about 150 hours on this course. More than half of that time is spent preparing for class and doing assignments. It's difficult to estimate how long it will take you to complete the course, but you can probably expect to spend an average of 3 or more hours working through each of the 38 sessions.

## Meet the Team

### Content Development

Haynes Miller is a Professor of Mathematics at MIT. In 2005 he was an MIT MacVicar Faculty Fellow in recognition of his outstanding contributions to undergraduate education. He has taught 18.03 many times and was the prime mover behind its current design. Professor Miller contributed many of the materials used in this OCW Scholar course. He was also the principal investigator behind the development of the Interactive Java® Demonstrations called Mathlets used here.

Dr. Jeremy Orloff is a lecturer in the Department of Mathematics and in the Experimental Study Group at MIT. He has taught 18.03 many times. Dr. Orloff was the lead content developer of this OCW Scholar course and worked closely with MIT OpenCourseWare on its development.

Dr. John Lewis is a Research Affiliate and former Senior Lecturer in the Department of Mathematics. He taught 18.03 for many years in the Experimental Study Group and Concourse programs at MIT, often in collaboration with Dr. Orloff.

### Video Lectures

Arthur Mattuck is an Emeritus Professor of Mathematics at MIT. He has been a major force in the design of undergraduate mathematics classes at MIT. Professor Mattuck taught 18.03 many times and his lecture videos and written notes are used throughout this OCW Scholar course.

### Teaching Assistants

- Dr. Lydia Bourouiba
- David Shirokoff

* To learn more about the Teaching Assistants, visit the **Meet the TAs** page.*

### Technical and Writing Assistance

- Heidi Burgiel
- John (Sweet Tea) Dorminy
- Shelby Heinecke
- Ailsa Keating
- Neil Olver
- Aviva Siegel
- Olga Stroilova
- Abiy Fekadu Tasissa