In mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. In other words, it singles out special operators, within the larger class of operators having variable coefficients. Such constant coefficient operators have been found to be the easiest to handle, in several respects. They include for example the Laplacian of potential theory and other major examples of mathematical physics. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In mathematics, a difference operator maps a function f(x) to another function f(x + a) − f(x + b). ... In mathematics a constant function is a function whose values do not vary and thus are constant. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... Potential theory may be defined as the study of harmonic functions. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ...

In the case of ordinary differential equations, writing In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an differential equation that contains functions of only one independent variable, and derivatives in that variable. ...

D = d/dx

the general constant-coefficient differential operator is

L = p(D),

where p is any polynomial with complex number coefficients. The solution of equations Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for one of the square roots of negative one (−1). ...

Lf = g

with a given function g(x) was given already in the eighteenth century, by Leonhard Euler. It has been suggested that Leonhard Euler/EB1911 biography be merged into this article or section. ...

For partial differential equations, the constant-coefficient operators are characterised geometrically by their translation invariance, and algebraically as polynomials in the partial derivatives. According to the Ehrenpreis-Malgrange theorem, they all have fundamental solutions. In mathematics, a partial differential equation (PDE) is an equation relating the partial derivatives of an unknown function of several variables. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. ... In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Greens function. ...