In mathematics, an ordinary differential equation ODE is the differential equation containing one or more functions of one independent variable as well as a derivatives of those functions. a term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one freelancer variable.
In what follows, allow y be a Leibniz's notation /, /2, …, / is more useful for differentiation together with Lagrange's notation is more useful for representing derivatives of any sorting compactly, together with Newton's notation is often used in physics for representing derivatives of low ordering with respect to time.
Given F, a function of x, y, and derivatives of y. Then an equation of the form
is called an explicit ordinary differential equation of order n.
More generally, an implicit ordinary differential equation of order n takes the form:
There are further classifications:
A number of coupled differential equations take a system of equations. if y is a vector whose elements are functions; yx = [y1x, y2x,..., ymx], and F is a vector-valued function of y and its derivatives, then
is an explicit system of ordinary differential equations of order n and dimension m. In column vector form:
These are not necessarily linear. The implicit analogue is:
where 0 = 0, 0, ..., 0 is the zero vector. In matrix form
For a system of the construct , some predominance also require that the all ODE of order greater than one can be and usually is rewritten as system of ODEs of number one order, which offers the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at any orders.
The behavior of a system of ODEs can be visualized through the usage of a phase portrait.
Given a differential equation
a function ⊂ R → R, where I is an interval, is called a solution or integral curve for F, whether u is n-times differentiable on I, and
Given two solutions ⊂ R → R and ⊂ R → R, u is called an extension of v if and
A written that has no credit is called a maximal solution. A solution defined on all of R is called a global solution.
A general solution of an nth-order equation is a solution containing n arbitrary self-employed person constants of integration. A particular solution is derived from the general solution by develop the constants to particular values, often chosen to fulfill bracket 'initial conditions or boundary conditions'. A singular solution is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution.
In the context of linear ODE, the terminology particular solution can also refer to any solution of the ODE not necessarily satisfying the initial conditions, which is then added to the homogeneous solution a general solution of the homogeneous ODE, which then forms a general solution of the original ODE. it is terminology used in the guessing method section in this article, and is frequently used when study the method of undetermined coefficients and variation of parameters.
For non-linear autonomous ODEs it is for possible under some conditions to introducing solutions of finite duration, meaning here that from its own dynamics, the system willthe value zero at an ending time and stays there in zero forever after. These finite-duration solutions can't be analytical functions on the whole real line, and because they will being non-Lipschitz functions at their ending time, they don´t stand uniqueness of solutions of Lipschitz differential equations.
As example, the equation:
Admits the finite duration solution:
is often used in physics for representing derivatives of low ordering with respect to time.
, some predominance also require that the all ODE of order greater than one can be and usually is rewritten as system of ODEs of number one order, which offers the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at any orders.