Differential-Transform

Chapter 1

Differential Equations

1.1] Ordinary Differential Equation

Ordinary differential equation has a wide role in mathematical discipline which is closely related to both pure mathematical research and real engineering world applications. Most mathematical formulations of physical laws are described in terms of ordinary and partial differential equations and this has been a great motivation for their study in the past. In the 20th century the extremely fast development of Science led to applications in the fields of chemistry, biology, medicine, population dynamics, genetic engineering, economy, social sciences and others. All these disciplines promoted to higher level and new discoveries were made with the help of this kind of mathematical modeling. At the same time real world problems have been and continue to be a great inspiration for pure mathematics. Particularly concerning ordinary differential equations they led to new mathematical models and challenged mathematicians and research engineers to look for new methods to solve them.

The attempt to solve physical problems led gradually to mathematical models involving an equation in which a function and its derivatives play important roles. However, the theoretical development of this new branch of mathematics Ordinary Differential Equation has its origins rooted in a small number of mathematical problems. These problems and their solutions led to an independent discipline with the solution of such equations an end in itself. According to some historians of mathematics the study of differential equations began in 1675 when Gottfried Wilhelm von Leibniz (1646-1716) wrote the equation