# Differential Equations

- Institution: Massachusetts Institute of Technology
- Department: Mathematics
- Taught by: Prof. Arthur Mattuck, Prof. Haynes Miller, Dr. Jeremy Orloff, Dr. John Lewis
- Term: Fall 2011
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [MIT OpenCourseWare](https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/)
- Licence: [CC BY-NC-SA 4.0](https://creativecommons.org/licenses/by-nc-sa/4.0/) (Non-commercial use only.)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

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 the equations and techniques most useful in science and engineering.

Course Format

This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials include:

Lecture Videos by Professor Arthur Mattuck.

Course Notes on every topic.

Practice Problems with Solutions.

Problem Solving Videos taught by experienced MIT Recitation Instructors.

Problem Sets to do on your own with Solutions to check your answers against when you’re done.

A selection of Interactive Java® Demonstrations called Mathlets to illustrate key concepts.

A full set of Exams with Solutions, including practice exams to help you prepare.

Content Development

Haynes Miller

Jeremy Orloff

Dr. John Lewis

Arthur Mattuck

## Lectures

1. [Introduction to the 18.03 Applets](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/d7f58f1c83e6), 9 min (Video)
2. [Separable Equations](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/060d7396a9c3), 8 min (Video)
3. [Geometric View of DE's: Direction Fields, Integral Curves](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/301503045acc) (Video)
4. [Example, Existence and Uniqueness](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/07c552b028d1) (Video)
5. [Example Where Existence and Uniqueness Fails](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/98e1b3f147bd) (Video)
6. [Exploration of the Isoclines Applet](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/6508f13af5ac), 11 min (Video)
7. [Direction Fields](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/03c5c7740ea2), 12 min (Video)
8. [Euler's Method](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/4bc2f44f2fcd) (Video)
9. [Example of Euler's Method](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/6cac71101755) (Video)
10. [Exploration of the Euler's Method Applet](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/3cae8cdf47ca), 7 min (Video)
11. [Better Methods](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/6c31549f19aa) (Video)
12. [Pitfalls](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/8db2cc25143b) (Video)
13. [Euler's Method](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/6125cd9ec1bb), 11 min (Video)
14. [Linear First Order ODE's: Definition](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/22071700f857) (Video)
15. [Linear First Order ODE's: Examples](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/e13799702d6f) (Video)
16. [Linear Equations](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/fddcf0f32197), 11 min (Video)
17. [Solutions to Linear First Order ODE's](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/3915d99e6b51) (Video)
18. [Solutions of First Order Linear Equations](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/613d335f6531), 9 min (Video)
19. [Complex Arithmetic, Euler's Formula](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/c8e078689363) (Video)
20. [Justification of Euler's Formula](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/321551823804) (Video)
21. [Polar Form](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/e2d5f586596c) (Video)
22. [Example: Complex Replacement](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/2cadd8b00257) (Video)
23. [Nth Roots](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/e0e6e555f115) (Video)
24. [Complex Numbers and Euler's Formula](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/36b9b03f7cdf), 12 min (Video)
25. [Three Proofs of the Basic Trigonometric Identity](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/4ad73f0b7a8f) (Video)
26. [Sinusoidal Functions](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/1a1f90c0e226), 16 min (Video)
27. [Solutions to Constant Coefficient Linear First Order ODE's](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/d4ffb33ca295) (Video)
28. [Examples of Constant Coefficient Linear First Order ODE's](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/4d33b0c0510d) (Video)
29. [More Examples of Constant Coefficient Equations](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/7fd836e9b715) (Video)
30. [First Order Constant Coefficient Linear ODE's](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/a52e6739e8af), 14 min (Video)
31. [Exploration of the Amplitude and Phase: First Order Applet](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/b2f6286e6de8), 9 min (Video)
32. [Sinusoidal Inputs](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/02f27a60151b), 14 min (Video)
33. [First Order Autonomous Equations](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/77b94dd5faf4) (Video)
34. [Bank Account](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/e665956e81e1) (Video)
35. [Example: Logistic Model](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/8614f5bf8332) (Video)
36. [Exploration of the Phase Lines Applet](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/962aef26cae3), 9 min (Video)
37. [Example: Logistic Model with Harvesting](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/171823f9b46c) (Video)
38. [Autonomous Equations and Phase Lines](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/63057d3c665b), 12 min (Video)
39. [Constant Coefficient Second Order Homogeneous DE's](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/5abf64ab6078) (Video)
40. [Real Roots: Overdamping](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/635b8b4694c3) (Video)
41. [Complex Roots: Underdamping](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/a40493177807) (Video)
42. [Repeated Roots: Critical Damping](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/8f6626bd2770) (Video)
43. [Real Solutions from Complex Roots](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/ce26d09e9a18) (Video)
44. [Homogeneous Constant Coefficient Equations: Real Roots](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/0221f321004c), 10 min (Video)
45. [Homogeneous Constant Coefficient Equations: Any Roots](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/4c2d3cc767dd), 12 min (Video)
46. [Damping and Pseudo-Frequency](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/ec5396324b1b) (Video)
47. [Exploration of the Damped Vibrations Applet](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/7027c0d4f8eb), 7 min (Video)
48. [Damped Harmonic Oscillators](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/699676da442f), 10 min (Video)
49. [Linear Operators and Superposition](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/3ce6695d8e3a), 51 min (Video)
50. [Inhomogeneous Second Order DE's: Definitions and Models](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/1992bcf452a1) (Video)
51. [Particular and General Solutions](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/f745a69dc4cf) (Video)
52. [Inhomogeneous DE's with Exponential Input](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/085cb86dd2eb) (Video)
53. [Forced Oscillations](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/f3d046875f44), 14 min (Video)
54. [Definition of Stability](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/79f56a7da258) (Video)
55. [Stability of Constant Coefficient Second Order DE's](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/da12a966c5e4) (Video)
56. [Gain and Phase Lag](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/cf88553b9b10), 11 min (Video)
57. [Undetermined Coefficients](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/5f702a7ad51c), 11 min (Video)
58. [Solving DE's with Exponential Input when p(a) = 0](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/cc9631c4330a) (Video)
59. [Resonance in the Undamped System](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/cef31f69b82e) (Video)
60. [Reconciling the Resonant and Non-resonant Solutions](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/4c57bd110d25) (Video)
61. [Pure Resonance](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/fe789362cbbf), 12 min (Video)
62. [Exploration of the Amplitude and Phase: Second Order II Applet](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/cd927928205b), 12 min (Video)
63. [Frequency Response](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/28f0b9358d3a), 15 min (Video)
64. [Introduction to Fourier Transform](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/15417787e82a) (Video)
65. [Fourier Series for Functions with Period 2L](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/61a286b3e7e0) (Video)
66. [Orthogonality Relations](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/5f32dd4b5081) (Video)
67. [Computing Fourier Series](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/3d67babe74b1), 15 min (Video)
68. [Even and Odd Functions](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/f6b4a7092852) (Video)
69. [Convergence of Fourier Series](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/40268ffac294) (Video)
70. [Exploration of the Fourier Coefficient Applet](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/4c5dfb2aee96), 11 min (Video)
71. [Manipulating Fourier Series](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/5e25f8a1dfd4), 15 min (Video)
72. [Solving ODE's, Resonant Frequencies](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/5d3d39c7c3e4) (Video)
73. [Hearing I](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/c66a12e80a88) (Video)
74. [Hearing II](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/63ba336d39fb) (Video)
75. [Linear ODE's with Periodic Input](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/efe42392f1c2), 12 min (Video)
76. [Step and Delta Functions: Integration and Generalized Derivatives](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/dec1f0a5fbd9), 10 min (Video)
77. [Unit Step and Impulse Response](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/0824a4b57a8c), 14 min (Video)
78. [Example: f(t)*1](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/7eca817de5c7) (Video)
79. [Example: Radioactive Dumping](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/15ae9c2b526c) (Video)
80. [Exploration of the Convolution Accumulation Applet](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/2824b8f76731), 7 min (Video)
81. [Convolution and Green's Formula](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/f9691668dc88), 10 min (Video)
82. [Laplace Transform](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/4c7da151d3eb) (Video)
83. [Notation and Linearity](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/049e50e346c8) (Video)
84. [Start of Laplace Table](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/84b63c683b3b) (Video)
85. [Functions of Exponential Type](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/fbaf214cc1d4) (Video)
86. [Laplace Transform of Powers](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/31732481e622) (Video)
87. [Laplace Transform: Basics](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/be34d924be34), 10 min (Video)
88. [Heaviside Coverup Method](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/e3674ed876c6) (Video)
89. [Partial Fractions and Laplace Inverse](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/87a7ef7e3352), 13 min (Video)
90. [Laplace: Solving ODE's](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/8a7d24b2ee6d), 12 min (Video)
91. [Laplace Transform of Convolution](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/d43d8e329ee0) (Video)
92. [Pole Diagrams](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/184eb046adb0), 11 min (Video)
93. [Introduction to Systems](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/18b2586e5bca) (Video)
94. [Autonomous Systems: Solutions are Parametrized Curves](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/f93e2e0e1367) (Video)
95. [Linear Systems of Equations](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/c8784618e0d7), 7 min (Video)
96. [Matrix Methods Introduction, First Example](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/493c055a09a2) (Video)
97. [Matrix Systems: General Terminology](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/b70237938b15) (Video)
98. [Complex Eigenvalues: Example](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/01784675eb51) (Video)
99. [Three Chamber Mixing Example: Repeated Roots](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/bfd7b17a6ca6) (Video)
100. [Linear Systems: Matrix Methods](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/5e753bd0ae2e), 8 min (Video)
101. [Linear Systems: Complex Roots](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/ba4ab44421d4), 12 min (Video)
102. [Sketching Solutions](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/0eb52d57c241) (Video)
103. [More Sketching Solutions](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/03afd205e818) (Video)
104. [Trace-Determinant Diagram](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/ab1d1a904d46), 19 min (Video)
105. [Phase Portraits](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/f46e73818139), 13 min (Video)
106. [Theory of Systems](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/cc87aa946d03) (Video)
107. [Fundamental Matrices](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/370621b31a7f) (Video)
108. [The Matrix Exponential I](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/d3de9fd760ce) (Video)
109. [The Matrix Exponential II](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/78934842d003) (Video)
110. [Example: Solving a System with the Matrix Exponential](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/e1bb26dd2589) (Video)
111. [Calculating the Exponential Matrix](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/27e48a138612) (Video)
112. [Variation of Parameters](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/87319e2ca759) (Video)
113. [Matrix Exponentials](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/dd1fe5f143fb), 12 min (Video)
114. [Decoupling](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/8eff101380be) (Video)
115. [The Relation Between Nonlinear Systems and First-order ODE's](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/294bac94d3bc) (Video)
116. [The Nonlinear Pendulum](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/d986aad4cf59) (Video)
117. [Example: Predator-Prey and Volterra's Principle](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/435609a22749) (Video)
118. [Linearization](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/7a632c1cd390), 16 min (Video)
119. [Limit Cycles](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/809e81c20ea7) (Video)
120. [Existence of Limit Cycles](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/8909083dc874) (Video)
121. [Non-existence of Limit Cycles](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/0af37cd285e9) (Video)
122. [A Story about Limit Cycles](https://invisiblenetwork.ai/learn/mit/18-03sc-differential-equations-fall-2011/7faead7599cc) (Video)