An ordinary differential equation is an equation containing a function of one independent variable and its derivatives. The derivatives are ordinary since partial derivatives apply only to functions of many independent variables. Linear differential equations have solutions that can be added and multiplied by coefficients. They are well-defined and understood with exact closed-form solutions. Ordinary differential equations that lack additive solutions are nonlinear. Therefore solving them is more intricate. Graphical and numerical methods applied may approximate solutions of ordinary differential equations and will yield useful information, often sufficing in the absence of exact, analytic solutions.
The notation for differentiation varies depending upon the author and upon which notation is more useful for the task at hand. The general definition of an ordinary differential equation lets F by a given function of x, y and derivatives of y. The equation is as followed
F (x, y, y’,…..y^(n-1)) = y^n
This is called an explicit ordinary differential equation of order n
More generally, an implicit ordinary differential equation of order n takes the form of
F (x, y, y’, y’’,……y^n) = 0
A differential equation not depending on x is called an autonomous. A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y.
A number of coupled differential equations form a system of equations. If y is a vector whose elements are functions; y(x) = [y1(x), y2(x),….,ym(x)]’ and F is a vector valued function of y and its derivatives.