the relationship between the displacement x and the time t of an object under the force F, is given by the differential equation which constrains the motion of a particle of constant mass m. In general, F is a function of the position x(t) of the particle at time t. The unknown function x(t) appears on both sides of the differential equation, and is indicated in the notation F(x(t)). Definitions In what follows, let y be a dependent variable and x an independent variable, and y = f(x) is an unknown function of x. The notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. In this context, the Leibniz's notation is more useful for differentiation and integration, whereas Lagrange's notation is more useful for representing derivatives of any order compactly, and Newton's notation is often used in physics for representing derivatives of low order with respect to time. General definition Given F, a function of x, y, and derivatives of y. Then an equation of the form is called an explicit ordinary differential equation of order n. More generally, an implicit ordinary differential equation of order n takes the form: There are further classifications: System of ODEs 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, then is an explicit system of ordinary differential equations of order n and dimension m. In column vector form: These are not necessarily linear. The implicit analogue is: where 0 = (0, 0, ..., 0) is the zero vector. In matrix form For a system of the form , some sources also require that the Jacobian matrix be non-singular in order to call this an implicit ODE [system]; an implicit ODE system satisfying this Jacobian non-singularity condition can be transformed into an explicit ODE system. In the same sources, implicit ODE systems with a singular Jacobian are termed differential algebraic equations (DAEs). This distinction is not merely one of terminology; DAEs have fundamentally different characteristics and are generally more involved to solve than (nonsingular) ODE systems. Presumably for additional derivatives, the Hessian matrix and so forth are also assumed non-singular according to this scheme, although note that any ODE of order greater than one can be