Solutions -- Polynomials -- More general equations -- Implicit equations -- Ordinary differential equations -- Existence and uniqueness : preliminaries -- Continuation -- First order equations -- separable equation : expanded -- Equilibria : a first look at stability -- Multiple equilibria -- linear equation -- Riccati equation -- Comparison sometimes helps -- Periodic solutions -- Differentiation in search of a solution -- Dependence on initial conditions -- Continuing continuation -- Insight not numbers -- Introduction -- Two simple schemes -- Key stuff -- RKF methods -- few examples -- Second order equations -- What is the initial value problem -- linear equation -- Approach 1 : dealing directly -- Approach 2 : the linear system -- Comments on the two approaches -- Constant coefficient equations : the homogeneous case -- What to do with solutions -- nonhomogeneous equation or the variation of parameters formula untangled -- Cosines and convolutions -- Second thoughts -- Linear and nonlinear systems -- Constant coefficient linear systems -- phase plane -- linear system and the phase plane -- Competition and predator-prey systems -- Harvesting -- conservative detour -- Stability and Gronwall-ing.