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) + … + 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 (λx, 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).