# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 22B: Differential Equations

**Approved:**2000-09-01 (revised 2024-05-03, Nachtergaele (2011);)

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

Search by ISBN on Amazon: 978-1119820512

**Prerequisites:**

**Suggested Schedule:**

Lecture(s) |
Sections |
Comments/Topics |

1 and 2 |
1.1-3 |
Introduction and terminology, direction fields, discussion and solution of some ODE |

3 |
2.1 |
Linear equations; integrating factors |

4 |
2.2 |
Separable equations |

5 |
2.3-4 |
Modeling, mechanics; Linear versus non-linear equations |

6 |
2.5 |
Autonomous equations; Population dynamics |

7 |
2.7 |
Numerical approximation; Euler’s method |

8 |
2.8 |
Existence and uniqueness theorem |

9 |
2.9 |
First order difference equations |

10 |
3.1 |
Homogeneous 2 |

11 and 12 |
3.2-3 |
Fundamental solutions, linear independence, Wronskian |

13 |
3.4 |
Complex roots |

14 |
3.5 |
Repeated roots; Reduction of order |

15 |
3.6 |
Nonhomogeneous equations; Method of undetermined coefficients |

16 |
3.7 |
Variation of parameters |

17 |
3.8-9 |
Applications to oscillating systems |

18 |
6.1 |
Laplace Transform, definition |

19 |
6.2 |
Solution of initial value problems with Laplace Transform |

20 |
7.1 |
Systems of linear ODE, introduction |

21 |
7.2-3 |
Review of related linear algebra |

22 |
7.4 |
Basic theory of first order linear systems |

23 |
7.5 |
Homogeneous linear systems with constant coefficients |

24 |
7.6 |
Complex eigenvalues |

25 |
7.7 |
Fundamental matrices |

26 |
7.8 |
Repeated eigenvalues |

27 |
7.9 |
Nonhomogeneous linear systems |

28 |
Applications and review |

**Additional Notes:**

This syllabus is based on 27 50-minute lectures. This usually leaves two lectures for midterms, e.g. midterm one covering the material of Chapters 1 and 2, and midterm two covering the material of Chapters 3 and 6. Alternatively, one can hold one midterm and have a lecture on applications (and/or review) at the end.

There are several interesting options to extend and/or modify this material:

- • One could spend additional time in the beginning on examples of different types of ODE, their solution and applications; then summarize the basic methods a bit more quickly.
- • Include treatment of linear systems using the Laplace transform.
- • Section 2.9 on first Order Difference Equations can be considered optional. Alternatively, it can be expanded with a discussion of iterated maps (e.g. the logistic map).
- • Section 2.7 on Numerical Approximations: Euler’s Method could be the starting point for project(s) using Matlab or another software package.
- • There are many computations in this course that can be further explored with a mathematical software package such as Matlab, Maple or Mathematica. In particular, Section 2.7 on Numerical Approximations: Eulerâs Method is the starting point for project(s) using Matlab or another software package.

**Learning Goals:**

- Identify and classify various types of differential equations.
- Solve first-order differential equations using analytical methods, including separation of variables and integrating factor.
- Compute numerical approximations of the solutions to differential equations using Euler’s method.
- Use geometric methods (slope field and phase lines) to qualitatively analyze first order differential equations.
- Determine whether unique solutions are guaranteed to exist.
- Use the characteristic equation to solve second-order linear differential equations with constant coefficients.
- Calculate solutions of second-order linear differential equations by the method of undetermined coefficients and variation of parameters.
- Represent solutions of a second-order linear differential equation in terms of a fundamental set of solutions.
- Use Laplace transforms to solve linear differential equations.
- Solve systems of linear differential equations using matrix methods and interpret the solutions.
- Construct differential equations that model processes in the natural world and solve the equations using appropriate methods.