Special Multiplicative Operators for the Solution of ODE – Invariants and Representations

The generalized multiplicative operator of differentiation is introduced in this paper. It is shown that the generalized multiplicative operator can be expressed as a product of two noncommutative but multiplicative exponential operators, though the generalized multiplicative operator is not an exponential operator itself. The generalized multiplicative operator is effectively exploited for the construction of solutions to nonlinear ordinary differential equations through formal transformations of invariants and representations of initial conditions. The concept of the generalized multiplicative operator provides the insight into the algebraic structure of solutions to nonlinear ordinary differential equations which cannot be identified using conventional exponential operators