Ordinary Differential Equation

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 n is an equation of the form


where y is a function of x, y^'=dy/dx is the first derivative with respect to x, and y^((n))=d^ny/dx^n is the nth derivative with respect to x.

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 n is said to be linear if it is of the form


A linear ODE where Q(x)=0 is said to be homogeneous. Confusingly, an ODE of the form


is also sometimes called "homogeneous."

In general, an nth-order ODE has n linearly independent solutions. Furthermore, any linear combination of linearly independent functions solutions is also a solution.

Simple theories exist for first-order (integrating factor) and second-order (Sturm-Liouville 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. 667-674) give canonical forms and solutions for second-order 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 Runge-Kutta 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 first-order ODE be given by


for i=1, ..., n and let the functions f_i(x_1,...,x_n,t), where i=1, ..., n, all be defined in a domain D of the (n+1)-dimensional space of the variables x_1, ..., x_n, t. Let these functions be continuous in D and have continuous first partial derivatives partialf_i/partialx_j for i=1, ..., n and j=1, ..., n in D. Let (x_1^0,...,x_n^0) be in D. Then there exists a solution of (4) given by


for t_0-delta<t<t_0+delta (where delta>0) satisfying the initial conditions


Furthermore, the solution is unique, so that if


is a second solution of (◇) for t_0-delta<t<t_0+delta satisfying (◇), then x_i(t)=x_i^*(t) for t_0-delta<t<t_0+delta. Because every nth-order ODE can be expressed as a system of n first-order ODEs, this theorem also applies to the single nth-order ODE.

An exact first-order ordinary differential equation is one of the form




An equation of the form (◇) with


is said to be nonexact. If


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


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


in (◇), it has a y-dependent integrating factor.

Other special first-order types include cross multiple equations


homogeneous equations


linear equations


and separable equations


Special classes of second-order ordinary differential equations include


(x missing) and


(y missing). A second-order linear homogeneous ODE


for which


can be transformed to one with constant coefficients.

The undamped equation of simple harmonic motion is


which becomes


when damped, and


when both forced and damped.

Systems with constant coefficients are of the form


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

Abel's differential equation


Airy differential equation


Anger differential equation


Baer differential equations


Bernoulli differential equation


Bessel differential equation


Binomial differential equation


Bôcher equation


Briot-Bouquet equation


Chebyshev differential equation


Clairaut's differential equation


Confluent hypergeometric differential equation


d'Alembert's equation


Duffing differential equation


Eckart differential equation


where eta=e^(deltax).

Emden-Fowler differential equation


Euler differential equation


Halm's differential equation


Hermite differential equation


Heun's differential equation


where w^'=dw/dx.

Hill's differential equation


Hypergeometric differential equation


Jacobi differential equation


Laguerre differential equation


Lamé's differential equation


where z^'=dz/dx.

Lane-Emden differential equation


Legendre differential equation


Linear constant coefficients


Lommel differential equation


Löwner's differential equation


Malmstén's differential equation


Mathieu differential equation


where V^'=dV/dv.

Modified Bessel differential equation


Modified spherical Bessel differential equation


where R^'=dR/dr

Rayleigh differential equation


Riccati differential equation


Riemann P-Differential Equation


where u^'=du/dz.

Sharpe's differential equation


Spherical Bessel differential equation


where R^'=dR/dr.

Struve differential equation


Sturm-Liouville equation


Gegenbauer differential equation


van der Pol equation


Weber differential equation


where y^'=dy/dz.

Whittaker differential equation


where u^'=du/dz.

See also

Adams' Method, First-Order Ordinary Differential Equation, Green's Function, Isocline, Laplace Transform, Leading Order Analysis, Majorant, Partial Differential Equation, Relaxation Methods, Runge-Kutta Method, Second-Order Ordinary Differential Equation, Simple Harmonic Motion, Undetermined Coefficients Method, Variation of Parameters Explore this topic in the MathWorld classroom

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Ordinary Differential Equation

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