ORDINARY DIFFERENTIAL EQUATIONS
The study of differential equations is such an extensive topic that even a brief survey of its methods and applications usually occupies a full course.
In mathematics, an ordinary differential equation (abbreviated ODE) is an equation containing a function of one independent variable and its derivatives. The derivatives are ordinary because partial derivatives only apply to functions of many independent variables.
The subject of ODEs is a sophisticated one (more so with PDEs), primarily due to the various forms the ODE can take and how they can be integrated. Linear differential equations are ones with solutions that can be added and multiplied by coefficients, and the theory of linear differential equations is well-defined and understood, and exact closed form solutions can be obtained. By contrast, ODEs which do not have additive solutions are non-linear, and finding the solutions is much more sophisticated because it is rarely possible to represent them by elementary functions in closed form — rather the exact (or "analytic") solutions are in series or integral form. As an alternative to the exact analytic solution, graphical and numerical methods (by hand or on computer) may be used to generate approximate solutions. The properties of such approximate solutions may yield very useful information, which often suffices in the absence of the exact analytic solution.
Ordinary differential equations (ODEs) arise in many different contexts throughout mathematics and science (social and natural) one way or another, because when describing changes mathematically, the most accurate way uses differentials and derivatives (related, though not quite the same). Since various differentials, derivatives, and functions become inevitably related to each other via equations, a differential equation is the result, describing dynamical phenomena, evolution and variation. Often, quantities are defined as the rate of change of other quantities (time derivatives), or gradients of quantities, which is how they enter differential equations.
Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), geology (weather modelling), chemistry (reaction rates), biology (infectious diseases, genetic variation), ecology and population modeling (population competition), economics (stock trends, interest rates and the market equilibrium price changes).
Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d’Alembert and Euler.