General Solution

general solution [¦jen·rəl sə′lü·shən]

(mathematics)

For annth-order differential equation, a function of the independent variables of the equation and ofnparameters such that assignment of any numerical values to the parameters yields a solution to the equation. Also known as general integral.

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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, C₁, …, C_n)

that are continuously dependent on n arbitrary constants C₁, …, 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. (SeeDIFFERENTIAL EQUATIONS.) If every function y defined by the relation ø(x, y, C₁, …, C_n) = 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 y’ = –x/y, the functions (upper semicircles) and y (lower semicircles) are general solutions; the relation x² + y₂ = C₂ (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.