In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations (PDEs) which may be with respect to more than one independent variable, and, less commonly, in contrast with stochastic differential equations (SDEs) where the modeled process is random.
Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
a 0 ( x ) y + a 1 ( x ) y ′ + a 2 ( x ) y ″ + ⋯ + a n ( x ) y ( n ) + b ( x ) = 0 , {\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''+\cdots +a_{n}(x)y^{(n)}+b(x)=0,}
where a 0 ( x ) , … , a n ( x ) {\displaystyle a_{0}(x),\ldots ,a_{n}(x)} and b ( x ) {\displaystyle b(x)} are functions that do not need to be linear, and
y ′ , … , y ( n ) {\displaystyle y',\ldots ,y^{(n)}} are the successive derivatives of the unknown function y {\displaystyle y} of the variable x {\displaystyle x} . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are solutions of linear differential equations (see Holonomic function). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example Riccati equation). Some ODEs can be solved explicitly in terms of known functions and integrals. When that is not possible, the equation for computing the Taylor series of the solutions may be useful. For applied problems, numerical methods for ordinary differential equations can supply an approximation of the solution.
Background Ordinary differential equations (ODEs) arise in many contexts of mathematics and social and natural sciences. Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related via equations, such that a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), 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), meteorology (weather modeling), 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. A simple example is Newton's second law of motion—the relationship between the displacement x {\displaystyle x} and the time t {\displaystyle t} of an object under the force F {\displaystyle F} , is given by the differential equation
m d 2 x ( t ) d t 2 = F ( x ( t ) ) {\displaystyle m{\frac {\mathrm {d} ^{2}x(t)}{\mathrm {d} t^{2}}}=F(x(t))\,}
which constrains the motion of a particle of constant mass m {\displaystyle m} . In general, F {\displaystyle F} is a function of the position x ( t ) {\displaystyle x(t)} of the particle at time t {\displaystyle t} . The unknown function x ( t ) {\displaystyle x(t)} appears on both sides of the differential equation, and is indicated in the notation F ( x ( t ) ) {\displaystyle F(x(t))} .
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