# 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 27B: Differential Equations with Applications to Biology

**Approved:**2018-03-01, S. Walcott, M. Goldman

**ATTENTION:**

Also named BIS 27B.

**Units/Lecture:**

Lecture—3 hour(s); Laboratory—2 hour(s).

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

"Fundamentals of Differential Equations," Ninth Edition by Nagle, Saff, and Snider

Search by ISBN on Amazon: 9780321977069

Search by ISBN on Amazon: 9780321977069

**Prerequisites:**

(MAT 027A C- or better or BIS 027A C- or better) or (MAT 022A C- or better, (MAT 022AL C- or better or ENG 006 C- or better or ECS 032A C- or better or ECS 036A C- or better or ECH 060 C- or better or EME 005 C- or better)).

**Course Description:**

Solutions of differential equations with biological, medical, and bioengineering applications.

**Suggested Schedule:**

Lecture | Section | Topics |
---|---|---|

1 | 1.1 | Background |

2 | 1.2 | Solutions and Initial Value Problems |

3 | 1.3 | Direction Fields |

4 | 2.2 | Separable Equations |

5 | 2.3 | Linear Equations (includes integrating factors) |

6 | 3.5 | Electrical Circuits |

7 | 3.1 3.2 |
Mathematical Modeling Population Dynamics |

8 | Strogatz 2.1-2.3 | Nonlinear Autonomous Equations (uses Pop dynamics example) |

9 | Strogatz 2.4 | One-dimensional Linear Stability Analysis |

10 | Exam | |

11 | 4.1 | Introduction: The Mass-Spring Oscillator |

12 | 4.2 | Homogeneous Linear Equations: The General Solution |

13 | 4.4 and 4.5 |
Nonhomogeneous Equations: The Method of Undetermined Coefficients The Superposition Principle and Undetermined Coefficients Revisited |

14 | 4.6 and 4.7 |
Variation of Parameters Variable-Coefficient Equations |

15 | 9.1-9.3 | Intro to Linear Systems, Linear Algebra Review |

16 | 9.4 (skim), 9.5 | Homogeneous Linear Systems with Constant Coefficients |

17 | 9.6 | Complex Eigenvalues |

18 | 9.7 | Nonhomogeneous Linear Systems |

19 | 9.8 | The Matrix Exponential Function & Fundamental Matrices |

20 | Exam | |

21 | Strogatz 6.1-6.2 | Introduction to the Phase Plane |

22 | Strogatz 6.3-6.4 | Multi-dimensional Linear Stability Analysis |

23 | (Notes) | Bifurcations in Nonlinear Systems |

24 | 7.2 and 7.3 |
Definition of the Laplace Transform Properties of the Laplace Transform |

25 | 7.4 and 7.5 |
Inverse Laplace Transform Solving Initial Value Problems |

26 | 7.8 | Convolution |

27 | 7.9 | Impulses and the Dirac Delta Function |

28 | Buffer lecture |

### Laboratory Content

Lab | Biology Example |
---|---|

1 | Logistic growth |

1 | Interaction between the immune system and HIV virus; exponential growth |

2 | Mechanical ventilation |

2 | Glucose concentration in blood (infusion) |

3 | Writing an Euler differential equation solver |

3 | Neural circuit |

3 | Leaky integrate-and-fire-neuron model |

4 | Writing a Runge-Kutta differential equation solver |

4 | Logistic growth; stability (move up to 1D stability) |

5 | Firefly synchronization |

5 | Human circadian rhythm |

5 | Repressilator model |

6 | Solving sytems of ODEs in Matlab |

6 | Morris-Lecar model |

7 | Frequency analysis/systems identification of muscle effector reponse (with Laplace and impulse response) |

8 | Genetic switch (move up: 1D) |

8 | Transcription kinetics |

9 | Run and tumble (diffusion equation) |

9 | Chemotaxis |

**Additional Notes:**

- 1.4 The Approximation Method of Euler is covered in lab
- 3.5 Electrical Circuits has a corresponding Neurons as Circuits lab
- 3.7 Higher-Order Numerical Methods: Taylor and Runge-Kutta - Runge-Kutta is covered in lab
- 10.1 Introduction: A Model for Heat Flow (Diffusion equation) is covered in final lab, as an example of how to numerically integrate a partial differential equation (if someone was on schedule and wanted to use the buffer lecture for this, they could optionally do 10.2 solve simple partial differential equation analytically using Method of Separation of Variables)