In This Paper We Examine the Nature of the Secondary Order Differential Equation Analytical Solutions With a State Derivative Formal Delay Considering a Convergent Power Series G(Z) of a Secondary Equation + With the Relation P(Z) + , We Obtain an Analytic Solution X(Z). Furthermore, We Characterize a Polynomial Solution When P(Z) Is a Polynomial. We Built a Corresponding Auxiliary Equation With Parameter to Obtain Analytical Solutions of the Problem. . the Existence of Solutions of an Auxillary Equation Depends on the Condition of a Parameter , Such As Is In the Unit Circle and Is a Root of Unity Which Satisfies the Diophantine Condition.