  # Differential Equations

### Course Description

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and eigenvectors; and Non-linear autonomous systems: critical point analysis and phase plane diagrams.

The original name of this course is: 18.03 Differential Equations.

Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (MIT OpenCourseWare: Massachusetts Institute of Technology), http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/Cours... License: Creative commons BY-NC-SA Linear Phase Portraits Mathlet from the d'Arbeloff Interactive Math Project. (Image courtesy of Hu Hohn and Prof. Haynes Miller.)
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### Video Lectures & Study Materials

Visit the official course website for more study materials: https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/

# Lecture Play Lecture
I. First-order Differential Equations
1 Integral Curves (48:54) Play Video
2 Euler's Method for y'=f(x,y) (50:43) Play Video
3 Solving First-order Linear ODE's (50:21) Play Video
4 Bernouilli and Homogeneous ODE's (50:11) Play Video
5 First-order Autonomous ODE's (45:44) Play Video
6 Complex Numbers and Exponentials (45:26) Play Video
7 First-order Linear with Constant Coefficients (41:09) Play Video
8 Applications (50:34) Play Video
II. Second-order Linear Equations
9 Solving Second-order Linear ODEs (49:58) Play Video
10 Undamped and Damped Oscillations (46:23) Play Video
11 Theory of General 2nd-Order ODEs (50:31) Play Video
12 Theory for Inhomogeneous ODE's (46:23) Play Video
13 Finding Sto Inhomogeneous ODE's (47:54) Play Video
14 Resonance (44:25) Play Video
III. Fourier Series
15 Introduction to Fourier Series (49:31) Play Video
16 Even and Odd Functions (49:27) Play Video
17 Solutions via Fourier Series (45:44) Play Video
IV. The Laplace Transform
19 Intro to the Laplace Transform (47:38) Play Video
20 Using the Laplace Transform (51:05) Play Video
21 Convolution Formula (44:18) Play Video
22 Using the Laplace Transform (44:07) Play Video
23 Dirac Delta Function (44:54) Play Video
V. First Order Systems
24 First-order Systems of ODE's (47:02) Play Video
25 Homogeneous Linear Systems (49:05) Play Video
26 Complex Eigenvalues (46:36) Play Video
27 Sketching Linear Systems (50:25) Play Video
28 Matrix Methods for Systems (48:52) Play Video
29 Matrix Exponentials (48:51) Play Video
30 Decoupling Linear System (47:05) Play Video
31 Non-linear Autonomous Systems (47:09) Play Video
32 Limit Cycles (45:52) Play Video
33 Relations Between Systems (50:08) Play Video