Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients
In this lesson, we learn how to sketch the solutions of a 2x2 homogeneous linear system with constant coefficients by modeling a sublimated war between two states. We start by solving the equations using a standard technique and then plotting the four easy solutions (pink, orange, blue, and green) that correspond to the normal modes. From there, we use the fact that the velocity field corresponding to the system of differential equations changes continuously to deduce that all trajectories are coming to the origin, but not necessarily in straight lines. This is an important skill to have when studying nonlinear equations.
Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients -- Lecture 27. Learn what graphs of systems of differential equations look like.
Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 28, 2008). License: Creative Commons BY-NC-SA.
More info at: http://ocw.mit.edu/terms
- Lecture Notes - Lecture Notes
- Supplementary Notes - Chapter 22 Supplementary Notes written by Prof. Miller
- What is a 2x2 Homogeneous Linear System?
- How do you solve a 2x2 Homogeneous Linear System?
- How do you sketch the solutions of a linear system of differential equations?
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Staff Review
A really interesting lecture with the always anecdotal professor Mattuck. You will learn how to truly understand the solutions to homogeneous linear systems with constant coefficients. You will actually see what is happening to a graph as t increases depending on what values of constants are.
