# General Solution

(redirected from*Weak solution*)

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## general solution

[¦jen·rəl sə′lü·shən]*n*th-order differential equation, a function of the independent variables of the equation and of

*n*parameters such that assignment of any numerical values to the parameters yields a solution to the equation. Also known as general integral.

## General Solution

A general solution of a differential equation

*y(n*) = *f(x, y, y*’, …, *y*^{(n–1)})

is a family of functions

*y* = ø(*x, C*_{1}, …, *C*_{n})

that are continuously dependent on *n* arbitrary constants *C*_{1}, …, *C _{n}* such that when these constants are appropriately selected, we can obtain any solution of the equation (a particular solution) unambiguously defined by the initial data, which fill a certain region of

*n*-dimensional space. (

*See*DIFFERENTIAL EQUATIONS.) If every function

*y*defined by the relation ø(

*x, y, C*

_{1}, …,

*C*) = 0 and satisfying the corresponding smoothness conditions is a general solution of the differential equation, then this relation is called a general integral of the differential equation. For example, for the differential equation

_{n}*y*’ = –

*x*/

*y*, the functions (upper semicircles) and

*y*(lower semicircles) are general solutions; the relation

*x*

^{2}+

*y*

_{2}=

*C*

_{2}(a family of circles) is a general integral (see Figure 1).

A general solution is similarly defined for a system of ordinary differential equations.

### REFERENCE

Stepanov, V. V.*Kurs differentsial’nykh uravnenii*, 8th ed. Moscow, 1959.