Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues (Real and Distinct Case)
In this lesson, we learn a more efficient way of solving systems of ODEs through matrices. The lecturer revisits a system of equations dealing with the chilling of a boiled egg, and instead of using elimination to solve it, he uses matrices. By reducing the problem of calculus of differential equations to solving algebraic equations, the lecturer reduces the calculus to algebra. The essential point is to view variables as not equal, and to focus on certain variables while demoting others to the background. By using the fundamental theorem of linear equations, the lecturer finds a condition on lambda, and from there, he is able to solve the system.
Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues (Real and Distinct Case) -- Lecture 25. Learn a more efficient way of solving systems of ODEs.
Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 27, 2008). License: Creative Commons BY-NC-SA.
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- Lecture Notes - Lecture Notes
- Supplementary Notes - Chapter 20 Supplementary Notes written by Prof. Miller
- Recitation Problems - Recitation Problems and classwork exercises
- Recitation Solutions - Recitation problem solutions
- How do you solve systems of differential equations?
- How do you solve homogeneous linear systems with constant coefficients?
- How do you use matrix eigenvalues to solve systems of differential equations?
- What is the superposition principle?
- What is a homogeneous system of differential equations?
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Staff Review
This lecture explains the more useful and practical way of solving systems of differential equations. Matrix Algebra, eigenvalues, and eigenvectors are used to solve them. Make sure you have a working knowledge of Linear Algebra for this lecture. A very interesting lecture with very real applications, in which multiple examples are actually solved.
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