The Introductory Differential Equations Companion is designed to assist students taking such a course through detailed examples that may be lacking in most textbooks due to space considerations. This book contains over a hundred examples found in a typical Introductory Differential Equations course at the collegiate level. Topics covered in this book include methods of solutions of first-ordered ordinary differential equations, linear and non-linear cases, including Separations of Variables, Integration Factors, Autonomic Equations, Bernoulli Equations, Numerical Methods, and applications in solving mixture problems and proportional growth problems. Further topics include solution methods for higher-ordered differential equations, homogenous and non-homogenous cases, Autonomic Equations with constant coefficients, Linear Independence and the Wronskian, the Method of Auxiliary Polynomials, Reduction of Order, Undetermined Coefficients and Cauchy-Euler Equations, with applications such as the bobbing spring-mass systems. Laplace Transforms are covered, with examples for solving non-homogenous initial-value problems using Laplace Transforms, with continuous and discontinuous (piecewise, impulse and periodic) forcing functions, and special cases. Solution methods using systems with eigenvalues and eigenvectors, both Real and Complex are discussed. Further topics include Variation of Parameters, Exact Equations and Series Solutions. A practice set with solutions is given to challenge the student with their skills.