Differential operator method is an effective method to solve the special solution of non-homogeneous linear differential equation with constant coefficients. Based on the operator polynomial theory, the special solution formula of the fractional operator of the second-order linear differential equation with constant coefficients with exponential function, trigonometric function, power function and their mixed functions is given. Examples show that the special solution formula is applicable, effective and simple in solving problems.

In mathematics, a differential operator is an operator defined as a function of differential operation. First of all, in notation, it is helpful to regard differentiation as an abstract operation, which accepts one function and obtains another function (in the form of higher-order function in computer science).

Application:

1. In the application of physical science, Laplace operator plays an important role in establishing and solving partial differential equations.

2. In differential topology, the outer derivative and Lie derivative operators have intrinsic meanings.

3. In abstract algebra, the concept of derivative is a generalization that differential operators need not be analyzed. Usually this generalization is used in algebraic geometry and commutative algebra.