The proposed book will introduce many physical applications of first-order ordinary differential equations, including biology, demography, economy, and finance. The theoretical framework is focused on the Cauchy problem and general solutions of homogeneous and nonhomogeneous second-order linear differential equations. The book also introduces second-order equations with applications to physics and mechanics and more advanced topics based on systems of first-order linear equations with applications to the spread of infections, cow rumen, prey, and predator modeling. The book will also contain topics on series solutions for first- and second-order linear equations, Laplace transform, and an introduction to stochastic differential equations and their applications in finance.