constantcoefficient

Constant coefficient refers to a coefficient in a mathematical expression that does not depend on the variables of the problem. In differential and difference equations, a linear equation is said to have constant coefficients if all coefficients of the derivatives, the function, and the shifts are constants. This is in contrast to variable-coefficient equations, where coefficients change with the independent variable or position.

In ordinary differential equations, constant coefficients enable solution by the characteristic equation. For example, the homogeneous equation y'' - 3y' + 2y = 0 has constant coefficients a = 1, b = -3, c = 2. The characteristic equation r^2 - 3r + 2 = 0 yields r = 1 and r = 2, giving the general solution y(x) = C1 e^x + C2 e^{2x}. More generally, equations with a constant coefficient p satisfy analogous exponential or polynomial-exponential solutions.

In linear difference equations, constant coefficients play a similar role. For a sequence defined by a_n = α a_{n-1} + β a_{n-2}, with constants α and β, the solution is found from the characteristic equation r^2 - α r - β = 0, and the form of the general solution is determined by the roots.

Constant coefficient operators extend to partial differential equations as well, such as (D^2 - 3D + 2)u = f, where D represents differentiation with respect to the independent variable. In practice, constant coefficients simplify analysis and yield closed-form solutions via exponentials, Laplace or Fourier transforms. Real-world problems often use constant-coefficient models as approximations when coefficients vary slowly or when a basic, solvable framework is desired.