# Homogeneous Equation

## homogeneous equation

[‚hä·mə′jē·nē·əs i′kwā·zhən]## 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

*a*_{0}(*x*)*y*^{(n)} + *a*_{1}(*x*)*y*^{(n-1)} + … + *a _{n}(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*) = *f* (λ*x*, λ*y*) 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*/(*x*^{2} + *y*^{2}).