ODEs tie an unknown function to derivatives in one variable
An ordinary differential equation links an unknown function of a single independent variable to that function’s derivatives. Newton’s second law written as mass times second derivative of position equals force is the textbook physics ODE. Some such equations yield exact formulas, while many others need series or numerical approximation.
An ordinary differential equation relates an unknown function of one independent variable to its derivatives, and the word 'ordinary' separates it from a partial differential equation. Its order is that of the highest derivative present. Newton's second law, linking the displacement of a body to the force acting on it through the second derivative, is the classic example, and the theory owes much to Euler, d'Alembert, Clairaut and Riccati as well as to Leibniz, several Bernoullis and Newton himself.
Such equations appear in geometry and analytical mechanics, celestial mechanics, weather modelling, chemical reaction rates, the spread of infectious diseases and genetic variation, competition between populations, and in economics for stock trends and interest rates. By the implicit function theorem most can, at least in theory, be rewritten in explicit or normal form, giving the highest derivative in terms of the lower ones. An equation is autonomous when it does not depend on the independent variable.
An equation counts as linear when the unknown function and its derivatives appear only to the first power and never multiplied together, with coefficients that may depend on x plus an extra term called the source, forcing term or inhomogeneity; when that term is present the equation is nonhomogeneous. Linear equations matter because most elementary and special functions in physics and applied mathematics solve them, and nonlinear models are generally approximated by linear ones.
Some equations can be solved explicitly with known functions and integrals. When that fails, the equation for the Taylor series of the solution may help, and in applied work numerical methods supply approximations. Several coupled equations form a system, which can be written as a single vector equation in which the n-th derivative of a vector of unknowns equals a function of x and the lower derivatives.
Source: Ordinary differential equation
