In this lesson, the focus is on solving systems of ODEs using fundamental matrices. The instructor reviews the properties of fundamental matrices, including the fact that they are not unique, but can be obtained by multiplying a given fundamental matrix by a non-singular square matrix of constants. The lesson then moves on to the application of matrix exponentials to solving systems, with the goal of finding a formula that does not require the calculation of eigenvalues or eigenvectors. The instructor shows how the formula for solving a one by one matrix can be generalized to solve the n by n case using the definition of the exponential function through an infinite series.
Matrix Exponentials; Application to Solving Systems -- Lecture 29. Solving systems of ODEs again; this time with some applications.
Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 29, 2008). License: Creative Commons BY-NC-SA.
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In this video, applications to systems of differential equations are discussed. Matrix exponentials (powers of matrices such as e^At and A^2 are included) and fundamental matrices are used to solve the systems.