Homogeneous Equation

homogeneous equation

[‚hä·mə′jē·nē·əs i′kwā·zhən]
(mathematics)
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.

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).

References in periodicals archive ?
To satisfy (8), we should be seeking [psi]([tau]) in such class of functions in which the homogeneous equation, corresponding (7), has two linearly independent solutions.
ln t/ln q]/[square root of ([mu](t))]I are linearly independent solutions of the homogeneous equation
i](t) are linear independent particular solutions to the homogeneous equation, [[gamma].
The solutions of the homogeneous equation are dilatational waves which are longitudinal waves, propagating along the direction of motion.
2] if and only if the coefficients satisfy the homogeneous equation given by
is the solution of the homogeneous equation u"(x) + u(x)= 0, that satisfies the initial conditions u(0) = 0, and u'(0) = 1, hence is the function [E.
Equation (2) is a homogeneous equation in an inhomogeneous medium.
h] is the general solution of the corresponding homogeneous equation
To get an idea about the minimal degree and the variation between different solutions, we first consider the underlying homogeneous equation
For any fixed k, the solution of the homogeneous equation is given by
2] system is likely to proceed according to a combination of heterogeneous Equation (4) and homogeneous Equation (8) pathways.