An ordinary differential equation (frequently called an "ODE," "diff eq," or "diffy Q") is an equality involving a function and its derivatives. An ODE of order is an equation of the form
(1)

where is a function of , is the first derivative with respect to , and is the th derivative with respect to .
Nonhomogeneous ordinary differential equations can be solved if the general solution to the homogenous version is known, in which case the undetermined coefficients method or variation of parameters can be used to find the particular solution.
Many ordinary differential equations can be solved exactly in the Wolfram Language using DSolve[eqn, y, x], and numerically using NDSolve[eqn, y, x, xmin, xmax].
An ODE of order is said to be linear if it is of the form
(2)

A linear ODE where is said to be homogeneous. Confusingly, an ODE of the form
(3)

is also sometimes called "homogeneous."
In general, an thorder ODE has linearly independent solutions. Furthermore, any linear combination of linearly independent functions solutions is also a solution.
Simple theories exist for firstorder (integrating factor) and secondorder (SturmLiouville theory) ordinary differential equations, and arbitrary ODEs with linear constant coefficients can be solved when they are of certain factorable forms. Integral transforms such as the Laplace transform can also be used to solve classes of linear ODEs. Morse and Feshbach (1953, pp. 667674) give canonical forms and solutions for secondorder ordinary differential equations.
While there are many general techniques for analytically solving classes of ODEs, the only practical solution technique for complicated equations is to use numerical methods (Milne 1970, Jeffreys and Jeffreys 1988). The most popular of these is the RungeKutta method, but many others have been developed, including the collocation method and Galerkin method. A vast amount of research and huge numbers of publications have been devoted to the numerical solution of differential equations, both ordinary and partial (PDEs) as a result of their importance in fields as diverse as physics, engineering, economics, and electronics.
The solutions to an ODE satisfy existence and uniqueness properties. These can be formally established by Picard's existence theorem for certain classes of ODEs. Let a system of firstorder ODE be given by
(4)

for , ..., and let the functions , where , ..., , all be defined in a domain of the dimensional space of the variables , ..., , . Let these functions be continuous in and have continuous first partial derivatives for , ..., and , ..., in . Let be in . Then there exists a solution of (4) given by
(5)

for (where ) satisfying the initial conditions
(6)

Furthermore, the solution is unique, so that if
(7)

is a second solution of (◇) for satisfying (◇), then for . Because every thorder ODE can be expressed as a system of firstorder ODEs, this theorem also applies to the single thorder ODE.
An exact firstorder ordinary differential equation is one of the form
(8)

where
(9)

An equation of the form (◇) with
(10)

is said to be nonexact. If
(11)

in (◇), it has an dependent integrating factor. If
(12)

in (◇), it has an dependent integrating factor. If
(13)

in (◇), it has a dependent integrating factor.
Other special firstorder types include cross multiple equations
(14)

homogeneous equations
(15)

linear equations
(16)

and separable equations
(17)

Special classes of secondorder ordinary differential equations include
(18)

( missing) and
(19)

( missing). A secondorder linear homogeneous ODE
(20)

for which
(21)

can be transformed to one with constant coefficients.
The undamped equation of simple harmonic motion is
(22)

which becomes
(23)

when damped, and
(24)

when both forced and damped.
Systems with constant coefficients are of the form
(25)

The following are examples of important ordinary differential equations which commonly arise in problems of mathematical physics.
(26)

(27)

(28)

(29)

(30)

(31)

Bernoulli differential equation
(32)

(33)

Binomial differential equation
(34)

(35)

(36)

Chebyshev differential equation
(37)

Clairaut's differential equation
(38)

Confluent hypergeometric differential equation
(39)

(40)

(41)

(42)

where .
EmdenFowler differential equation
(43)

(44)

(45)

(46)

(47)

where .
(48)

Hypergeometric differential equation
(49)

(50)

Laguerre differential equation
(51)

(52)

where .
LaneEmden differential equation
(53)

Legendre differential equation
(54)

(55)

(56)

Löwner's differential equation
(57)

Malmstén's differential equation
(58)

(59)

where .
Modified Bessel differential equation
(60)

Modified spherical Bessel differential equation
(61)

where
Rayleigh differential equation
(62)

(63)

Riemann PDifferential Equation
(64)

where .
Sharpe's differential equation
(65)

Spherical Bessel differential equation
(66)

where .
(67)

(68)

Gegenbauer differential equation
(69)

(70)

(71)

where .
Whittaker differential equation
(72)

where .