nonhomogeneous

Nonhomogeneous is a descriptor used across mathematics, physics, and engineering to denote systems in which a property is not uniform or not solely dependent on the internal structure. In mathematics, it often refers to equations or problems that include an external term or spatially varying coefficients, in contrast to a homogeneous case that is entirely self-contained.

In linear differential equations, a problem of the form L[y] = f(x) is nonhomogeneous when the forcing term f(x) is not zero. The corresponding homogeneous problem L[y] = 0 has solutions called the complementary or homogeneous solutions. The general solution to the nonhomogeneous equation is the sum of a homogeneous solution y_h and a particular solution y_p that satisfies L[y_p] = f(x). Common methods to obtain y_p include the method of undetermined coefficients (for certain constant-coefficient cases) and variation of parameters. For linear partial differential equations, nonhomogeneous terms represent sources or forcing terms, and techniques such as Green’s functions or Fourier methods help construct solutions.

Nonhomogeneous also appears in other contexts. In materials science and physics, a nonhomogeneous medium has properties that vary with position, such as density, elasticity, or refractive index. This spatial variation leads to models with position-dependent coefficients and typically more complex behavior of waves, diffusion, or other processes.

Example: the equation y'' − 3y' + 2y = e^x is nonhomogeneous because the right-hand side is nonzero. A corresponding particular solution is y_p = −x e^x, giving a full solution that combines y_h (from the homogeneous equation y'' − 3y' + 2y = 0) with y_p.

Overall, nonhomogeneous problems introduce external or spatially varying influences that break uniformity, requiring distinct solution techniques and interpretations.