nonhomogene

Nonhomogeneous, often called nonhomogeneous or non-homogeneous, describes equations or systems that include terms preventing simple scaling symmetry. In contrast, a homogeneous equation has no external forcing term and its solutions form a vector space closed under addition and scalar multiplication.

In differential equations, a linear nonhomogeneous equation has the form L[y] = g(t) with g(t) not identically zero. The related homogeneous equation L[y] = 0 defines the complementary solution y_c, and the general solution is y = y_c + y_p, where y_p is a particular solution. Example: y' - y = t. The homogeneous part gives y_h = C e^{t}. A particular solution is y_p = -t - 1, so the general solution is y = C e^{t} - t - 1.

In partial differential equations, nonhomogeneity often arises from sources or forcing terms. For instance, Poisson's equation ∇^2 φ = ρ is nonhomogeneous when ρ ≠ 0, while Laplace's equation ∇^2 φ = 0 is homogeneous.

In physics and engineering, nonhomogeneous models describe systems with external inputs or spatially varying properties. Variable coefficients and source terms lead to nonhomogeneous behavior and require different analytical or numerical techniques than the homogeneous case.

Methods for solving include the method of undetermined coefficients and variation of parameters for ODEs, and Green's functions for PDEs. In practice, numerical methods such as finite difference or finite element methods are common for more complex or irregular problems.

Applications of nonhomogeneous models span heat conduction with internal sources, forced vibrations, electrical circuits with external inputs, and materials with nonuniform properties.