Ordinary Differential Equations

Basic notions and definitions for ordinary differential equations of first order (linear and nonlinear equations general solutions, solutions of initial value problems, specific solutions, stationary solutions and stability). First order equations (separable, linear, Bernoulli, exact) and applications in population growth models, epidemiology and physics. The existence and uniqueness theorem. Linear second order equations with constant coefficients: Methods for solving homogeneous and non-homogeneous equations, equations with non-constant coefficients, Euler’s equation. Oscillations and other applications from the natural sciences. Series solutions for ordinary differential equations. Laplace transform for second order equations and the case of discontinuous forcing. Systems of linear equations: The matrix approach to linear systems-eigenvalues and eigenvectors. Introduction to dynamical systems: The flow of the 1st order differential equation. Ordinary differential equations and scientific software. The phase plane of linear systems and illustration of representative examples for nonlinear systems and their significance.