Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable.

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
:$a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0,$
where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable .

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'' and the time ''t'' of an object under the force ''F'', is given by the differential equation

:$m \frac = F(x(t))\,$

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 $(\dot y, \ddot y, \overset)$ 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

:$F\left (x,y,y',\ldots, y^ \right )=y^$

is called an ''explicit ordinary differential equation of order n''.

More generally, an '' implicit'' ordinary differential equation of order ''n'' takes the form:

:$F\left(x, y, y', y'',\ \ldots,\ y^\right) = 0$

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'') = 'y''₁(''x''), ''y''₂(''x''),..., ''y_m''(''x'') and F is a vector-valued function of y and its derivatives, then

:$\mathbf^ = \mathbf\left(x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^ \right)$

is an ''explicit system of ordinary differential equations'' of ''order'' ''n'' and ''dimension'' ''m''. In column vector form:

:$\begin y_1^ \\ y_2^ \\ \vdots \\ y_m^ \end = \begin f_1 \left (x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^ \right ) \\ f_2 \left (x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^ \right ) \\ \vdots \\ f_m \left (x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^ \right) \end$

These are not necessarily linear. The ''implicit'' analogue is:

:$\mathbf \left(x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^ \right) = \boldsymbol$

where 0 = (0, 0, ..., 0) is the zero vector. In matrix form

:$\begin f_1(x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^) \\ f_2(x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^) \\ \vdots \\ f_m(x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^) \end=\begin 0\\ 0\\ \vdots\\ 0 \end$

For a system of the form $\mathbf \left(x,\mathbf,\mathbf'\right) = \boldsymbol$, some sources also require that the Jacobian matrix $\frac$ be non-singular in order to call this an implicit ODE ystem 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 (and usually is) rewritten as system of ODEs of first order, which makes the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders.

The behavior of a system of ODEs can be visualized through the use of a phase portrait.

Solutions

Given a differential equation
:$F\left(x, y, y', \ldots, y^ \right) = 0$
a function , where ''I'' is an interval, is called a ''solution'' or integral curve for ''F'', if ''u'' is ''n''-times differentiable on ''I'', and
:$F(x,u,u',\ \ldots,\ u^)=0 \quad x \in I.$

Given two solutions and , ''u'' is called an ''extension'' of ''v'' if and
:$u(x) = v(x) \quad x \in I.\,$
A solution that has no extension is called a ''maximal solution''. A solution defined on all of R is called a ''global solution''.

A ''general solution'' of an ''n''th-order equation is a solution containing ''n'' arbitrary independent constants of integration