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.

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 ?
The introduction of ideal gas law into term 1 will yield a dimensional homogeneous equation for gas permeability.
is the solution to the associated homogeneous equation (obtained by setting f = 0 in (3.6)) and
To find the complementary solution of (21), consider the solutions [mathematical expression not reproducible] for the homogeneous equation
A general solution [mathematical expression not reproducible] of the homogeneous equation, corresponding to the nonhomogeneous equation (1) with respect to u(x), is represented as follows [18]:
The homogeneous equation AX = [O.sub.m,n] has a nontrivial symmetric solution.
Since [[z.sup.ln t/l nq]/[square root of [mu](t)]I and [[z.sup.-ln t/ln q]/[square root of [mu](t)] are linearly independent solutions of the homogeneous equation
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.
The corresponding homogeneous equation AXB + [CX.sup.T]D = 0 has a unique zero solution X = 0.
where [[phi].sub.i](t) are linear independent particular solutions to the homogeneous equation, [[gamma].sub.j](t) will be determined in such a way that after inserting the expression (38) into (36) we obtaint the identity.
The solutions of the homogeneous equation are dilatational waves which are longitudinal waves, propagating along the direction of motion.
is self-integrable in the rectangular region R bounded the lines x = [l.sub.1], y = [l.sub.2], x = [u.sub.1] and y = [u.sub.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.sub.1] (x) = sin x, [for all] x [member of] (0, [infinity]).

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