information:filtering
Solutions

Given a differential equation

a function u: I ⊂ R → R, where I is an interval, is called a solution or integral curve for F, if u is n -times differentiable on I, and

Given two solutions u: J ⊂ R → R and v: I ⊂ R → R, u is called an extension of v if I ⊂ J and

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. A particular solution is derived from the general solution by setting the constants to particular values, often chosen to fulfill set ' 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. This is the terminology used in the guessing method section in this article, and is frequently used when discussing the method of undetermined coefficients and variation of parameters.