Lectures (Video)

- 1. The Geometrical View
- 2. Euler's Numerical Method
- 3. Solving First-order Linear ODE's
- 4. First-order Substitution Methods
- 5. First-order Autonomous ODE's
- 6. Complex Numbers
- 7. First-order Linear ODEs with Constant Coefficients
- 8. Applications of First-order Linear ODEs with Constant Coefficients
- 9. Solving Second-order Linear ODE's with Constant Coefficients
- 10. Complex Characteristic Roots
- 11. Theory of General Second-order Linear Homogeneous ODE's
- 12. Stability Criteria
- 13. Particular Solution to Inhomogeneous ODE's
- 14. Resonance
- 15. Introduction to Fourier Series
- 16. More General Periods
- 17. Finding Particular Solutions via Fourier Series
- 19. Introduction to the Laplace Transform
- 20. Derivative Formulas
- 21. Convolution Formula
- 22. Using Laplace Transform to Solve ODEs
- 23. Impulse Inputs
- 24. First-order Systems of ODEs
- 25. Homogeneous Linear Systems
- 26. Repeated Real Eigenvalues
- 27. Sketching Solutions of Homogeneous Linear System
- 28. Matrix Methods for Inhomogeneous Systems
- 29. Matrix Exponentials
- 30. Decoupling Linear Systems
- 31. Non-linear Autonomous Systems
- 32. Limit Cycles
- 33. Relation Between Non-linear Systems and First-order ODEs

## Differential Equations

### Course Summary

*18.03 Differential Equations, Spring 2006*made available by

*Massachusetts Institute of Technology: MIT OpenCourseWare*under the

*Creative Commons BY-NC-SA*license.

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.

These video lectures by Professor Arthur Mattuck were recorded live in the Spring 2003 and do not correspond precisely to the lectures taught in the Spring of 2006. Professor Mattuck has inspired and informed generations of MIT students with his engaging lectures.

### Reading Material

1.**Textbook (MIT 18.03): Elementary Differential Equations with Boundary Value Problems. 5th ed.**

Edwards, C., and D. Penney. Elementary Differential Equations with Boundary Value Problems. 5th ed. Upper Saddle River, NJ: Prentice Hall, 2003. ISBN: 013145773X.

### Course Material

1.**Recitation Questions (MIT 18.03)**

TOPICS | SOLUTIONS | |
---|---|---|

1 | Natural Growth and Decay (PDF) | (PDF) |

2 | Direction Fields, Integral Curves, Isoclines (PDF) | (PDF) |

3 | Numerical Methods; Linear Models (PDF) | (PDF) |

4 | First Order Linear ODEs: Models and Solutions (PDF) | (PDF) |

5 | Complex Numbers, Complex Exponentials (PDF) | (PDF) |

6 | Using the Complex Exponential; Autonomous Equations (PDF) | (PDF) |

7 | Solutions to Second Order ODEs (PDF) | (PDF) |

8 | Homogeneous Second Order Linear Equations (PDF) | (PDF) |

9 | Second Order Linear Equations (PDF) | (PDF) |

10 | Operators, Exponential Response, Exponential Shift, Undetermined Coefficients (PDF) | (PDF) |

11 | Superposition, Frequency Response (PDF) | (PDF) |

12 | Review | |

13 | Fourier Series: Introduction (PDF) | (PDF) |

14 | Fourier Series: Playing Around (PDF) | (PDF) |

15 | Fourier Series: Harmonic Response (PDF) | (PDF) |

16 | Step and Delta Functions, and Step and Delta Responses (PDF) | (PDF) |

17 | Convolution (PDF) | (PDF) |

18 | Laplace Transform (PDF) | (PDF) |

19 | Hour Exam Review (PDF) | (PDF) |

20 | Systems of First Order Equations (PDF) | (PDF) |

21 | Eigenvalues and Eigenvectors (PDF) | (PDF) |

22 | Complex or Repeated Eigenvalues (PDF) | (PDF) |

23 | Qualitative Analysis of Linear Systems (PDF) | (PDF) |

24 | Matrix Exponentials and Inhomogeneous Equations (PDF) | (PDF) |

25 | Qualitative Analysis of Nonlinear Systems (PDF) | (PDF) |

26 | Review |

2.

**Assignments (MIT 18.03)**

LEC # | ASSIGNMENTS | SOLUTIONS |
---|---|---|

4 | Problem Set 1 (PDF) | (PDF) |

8 | Problem Set 2 (PDF) | (PDF) |

14 | Problem Set 3 (PDF) | (PDF) |

17 | Problem Set 4 (PDF) | (PDF) |

23 | Problem Set 5 (PDF) | (PDF) |

26 | Problem Set 6 (PDF) | (PDF) |

28 | Problem Set 7 (PDF) | (PDF) |

34 | Problem Set 8 (PDF) | (PDF) |

37 | Problem Set 9 (PDF) | (PDF) |

3.

**Exam questions and solutions (MIT 18.03)**

PRACTICE EXAMS | EXAMS | SOLUTIONS | |
---|---|---|---|

L10 | Practice Exam and Solutions (PDF) | (PDF) | (PDF) |

L19 | Study Guide and Two Practice Exams with Solutions (PDF) | (PDF) | (PDF) |

L30 | Formula Sheet and Two Practice Exams with Solutions (PDF) | (PDF) | (PDF) |

L40 | Formula Sheet and Two Practice Final Exams With Solutions (PDF) | (PDF) |

### Other Resources

Not available.### Software

1.**Mathlets tool**

Ordinary Differential Equations Manipulatives software tool.