Consider the following systems of linear equations. For convenience, the solutions are given. Express an arbitrary solution as the sum of an element of the kernel of the transformation defined by the matrix of coefficients and a particular solution. X1 - X2 + X3 = 3 -2x1 + 2x2 - 2x3 = -6 Solutions: x1 = r - s + 3, x2 = r, x3 = s

FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS III: Applications and More Analytic Methods David Levermore Department of Mathematics University of Maryland 19 February 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 8. First-Order Equations: Applications 8.1. General Guidelines 2 8.2. Tanks and Mixtures 2 8.3. Loans