This course introduces the basic theories and methods in ordinary differential equations (ODEs) and lots of examples to illustrate the application of ODEs in physics, chemistry, biology, etc. The course aims to nurture the ability of the student to use ODE to solve problems arising in applications. The theoretical part of the course covers: first order linear equations, separation of variables, direction fields, Euler’s method, existence and uniqueness theorem, phase line analysis, second order linear equations, variation of constants, asymptotic behavior of solutions, Laplace transform, first order linear systems, first order nonlinear autonomous systems, linear stability and types of equilibria, phase plane analysis, nullclines, invariant regions, etc.