Syllabus Detail

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
Prerequisites:
Prerequisite(s): BIS 027A C- or better or (MAT 022A C- or better, (MAT 022AL C- or better or ENG 006 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

LabBiology Example
1Logistic growth
1Interaction between the immune system and HIV virus; exponential growth
2Mechanical ventilation
2Glucose concentration in blood (infusion)
3Writing an Euler differential equation solver
3Neural circuit
3Leaky integrate-and-fire-neuron model
4Writing a Runge-Kutta differential equation solver
4Logistic growth; stability (move up to 1D stability)
5Firefly synchronization
5Human circadian rhythm
5Repressilator model
6Solving sytems of ODEs in Matlab
6Morris-Lecar model
7Frequency analysis/systems identification of muscle effector reponse (with Laplace and impulse response)
8Genetic switch (move up: 1D)
8Transcription kinetics
9Run and tumble (diffusion equation)
9Chemotaxis
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)