The matrix exponential is a powerful computational and conceptual tool for analyzing systems of linear, constant coefficient, ordinary differential equations (ODE's). This narrative offers a quick introduction to the technique, with examples and exercises. It also includes an introduction to Putzer's Algorithm for computing the matrix exponential, a topic not usually covered in introductory ODE courses. This simple algorithm allows one to compute the exponential of any matrix whose eigenvalues are known. A computer algebra system can be helpful for this material, since the computations can be a bit cumbersome. For the reader who wants to know a bit more, proofs for many results are included in an appendix. The analysis, however, relies only on elementary matrix algebra and techniques for series that most students see in a second semester calculus course.
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