In each case, find an elementary matrix E that satisfies the given equation. EA B

FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Linear Systems David Levermore Department of Mathematics University of Maryland 23 April 2012 Because the presentation of this material in lecture will diﬀer from that in the book, I felt that notes that closely follow the lecture presentation might be appreciated. Contents 1. Introduction 1.1. Normal Form and Solutions 2 1.2. Initial-Value Problems 3 1.3. Recasting Higher-Order Problems as First-Order Systems 3 1.4. Numerical Methods 5 2. Linear Systems: General Methods and Theory 2.1. Initial-Value Problems 7 2.2. Homogeneous Systems 8 2.3. Wronskians and Fundamental Matrices 9 2.4. Natural Fundamental Matrices 12 2.5. Nonhomogenous Systems and Green Matrices (not covered) 14 B. Appendix: Vectors and Matrices B.1. Vector and Matrix Operations 17 B.2. Invertibility and Inverses 29