Homogeneous Equation

homogeneous equation

[‚hä·mə′jē·nē·əs i′kwā·zhən]
An equation that can be rewritten into the form having zero on one side of the equal sign and a homogeneous function of all the variables on the other side.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Homogeneous Equation


an equation whose form does not change upon simultaneous multiplication of all or only some unknowns by a given arbitrary number. In the latter case, the equation is said to be homogeneous with respect to the corresponding unknowns. For example, xy + yz + zx = 0 is a homogeneous equation with respect to all unknowns, and the equation y + ln (x/z) + 5 = 0 is homogeneous with respect to x and z. The left-hand member of a homogeneous equation is a homogeneous function. The equation

a0(x)y(n) + a1(x)y(n-1) + … + an(x)y = 0

which is called a linear homogeneous differential equation, is homogeneous with respect to y, y′, …,y(n-1), y(n). The equation y′ = f(x, y), where f(x, y) = fx, λy) for any λ [f(x, y) is a homogeneous function with a degree of homogeneity 0], is said to be a differentia) equation homogeneous with respect to the variables x and y. For example, y′ = xy/(x2 + y2).

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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