write the equation of the line passingthrough P with direction vector d in (a) vector form and(b) parametric form. 6.P 11, 02, d c d 13d

HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Homogeneous Equations David Levermore Department of Mathematics University of Maryland 12 March 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. 1. Introduction 1.1. Normal Form and Solutions 2 1.2. Initial Value Problems 2 1.3. Intervals of Deﬁnition 4 2. Homogeneous Equations: General Methods and Theory 2.1. Method of Linear Superposition 5 2.2. Wronskians 9 2.3. Fundamental Sets of Solutions and General Solutions 10 2.4. Natural Fundamental Sets of Solutions 12 2.5. Linear Independence of Solutions 16 2.6. Method of Order Reduction 19 3. Homogeneous Equations with Constant Coeﬃcients 3.1. Characteristic Polynomials and the Key Identity 22 3.2. Real Roots of Characteristic Polynomials 23 3.3. Complex Roots of Characteristic Polynomials