This course is an introduction to ordinary differential equations. We study a variety of models in physics and engineering, which use differential equations, we learn how to construct differential equations, corresponding to different ecological or physical systems. In addition to introducing various analytical techniques for solving basic types of ODEs, we also study qualitative techniques for understanding the behavior of solutions. We describe a collection of methods and techniques used to find solutions to several types of differential equations, including first order scalar equations, second order linear equations, and systems of linear equations. We introduce Laplace transform methods to solve constant coefficients equations with generalized source functions. We also provide a brief introduction to boundary value problems, Sturm-Liouville problems, and Fourier Series. Near the end of the course, we combine previous ideas to solve an initial boundary value problem for a particular partial differential equation, the heat propagation equation.