Compute u + v. u - v. 2u. and 3u - 2v if (, ) ~[;] , ~ [-;J (h, ~n ~m , ~m,~m

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 5 8.4. Motion 6 8.5. Population Dynamics 9 9. Exact Diﬀerential Forms and Integrating Factors 9.1. Implicit General Solutions 12 9.2. Exact Diﬀerential Forms 13 9.3. Integrating Factors 17 10. Special First-Order Equations and Substitution 10.1. Linear Argument Equations (not covered) 21 10.2. Dilation Invariant Equations (not covered) 22 10.3. Berno