[MUSIC] Hello everyone. Let's start the course on ordinary differential equations. Even though we can catch a very small tip of this huge iceberg through this introductory course, I do hope that you can enjoy some beauty and the usefulness of the theory of ordinary differential equations. First, let me introduce some basic definitions and terminologies related to the theory of differential equations. There will be the common words which we are going to use throughout the course. Are you ready? So let's get into the chapter one introduction, okay? First the basic definitions and terminologies, okay? The mathematical modeling of our physical, or biological, or the ecological phenomenon, often produces an equation, which involves ordinary or partial derivatives of some unknown function, okay? Such an equation is called a differential equation. There are two types of differential equations. First, the differential equation involving only ordinary derivative respect to a single independent variable is called an ordinary differential equation. On the other hand, a differential equation involving partial derivatives with respect to more than one independent variable is called a partial differential equation. For example, Newton's second law of motion applied to a free falling body leads to an ordinary differential equation. Say, m times d squared h over d t squared is equal to m times dv over dt is equal to negative m times g. Where m is the mass of the object, g the gravitational acceleration, h the height of the object, and v which is equal to dh over dt, is the velocity. On the other hand, modeling of vibrating strings, under some ideal conditions, leads to a partial differential equation called the wave equation, which is d square u over d u squared minus c square times d square u over dx square is equal to zero. Where t is the time, x the location along the string, c the wave speed, and u as a function of x, and time is the displacement of the string.