This course on Linear Systems of Differential Equations aims to teach students the fundamentals of using linear algebra to analyze and solve differential equations. By the end of the course, students will be able to identify equilibrium points, derive straight line solutions, understand linear independence among vectors, and find general solutions using vectors. The teaching method involves a combination of theoretical explanations, matrix notations, vector practices, and visualizations in the phase plane. This course is intended for individuals interested in mathematics, particularly those looking to deepen their understanding of differential equations and linear algebra concepts.
Overview
Syllabus
Dr. Devaney Reintroduces the Mass Spring System
Using Linear Algebra as a Method
Review Matrix Notation With an Example
Practice Vector Notation for Linear Systems
Always Begin by Finding Equilibrium Points
All Linear System Origins Have an Equilibrium Point
Derive the Important Straight Line Solution
Discover a Property of Constants
Examine the Solutions in the Phase Plane
Two Solutions Reveal All in Linear Algebra
Devaney Reveals the Linearity Principle
Combine Solutions to Find a General Solution
The Phase Plane Visualizes the Alternate Solutions
Alternate Solutions Lead to a Saddle Phase Plane
Vectors Reveal a Quicker Route to a General Solution
Linear Independence Among Vectors
Linear Independence Leads to Constant Solutions
Finding Homogeneous and Non-Homogeneous Solutions
Straight Line Solutions Show Decaying Exponentials
Taught by
Wondrium
