Matrix Science Mathematic (MSMK)

A new classification of functions is presented in this work in relation to ordinary differential equations. A function for which the n’th derivative of it can be expressed in terms of linear combinations of lower derivatives and the function itself is defined as the linear differential function where n is the lowest derivative for such a relation to hold. The functions that do not obey the rule are defined as nonlinear differential functions. The properties of the differential functions are discussed through theorems and examples. One elementary application is the method of undetermined coefficients where the non-homogenous function should be a linear differential function. Variable coefficient equations are also treated within the context of linear differential functions. The approximation of nonlinear differential functions in terms of linear differential functions are discussed. An application to the perturbation solutions is given. The ideas and definitions presented in this work will add to the understanding of differential equations and their solutions as functions.