This paper introduces generalized trigonometric and hyperbolic functions defined by exponential representations involving cube roots of -1. Through the illustration of their definitions, proofs, and properties, these functions are explained as solutions to certain non-homogeneous equations. A characterization theorem is presented, demonstrating uniqueness based on specified conditions. The relationship between roots of generalized trigonometric functions and hyperbolic functions are examined. Furthermore, the application of these functions in solving a differential equation is demonstrated.