**ABSTRACT**

This paper presents block multi-derivative linear multi-step method of single block to solve initial value problems of first and second order ordinary differential equations. The derivation of the method is achieved via interpolation and collocation approaches using power series polynomial as basis function. Higher derivative terms are introduced to improve the order of accuracy of the method and also give room to modify the method for solving initial value problems (IVPs) of second-order ordinary differential equations (ODEs). The fundamental properties of the method were carried out and they are found to be zero stable and consistent, hence the methods are convergent. The method is implemented in block mode to solve mathematical equations resulting from modeling Artificial Intelligent (AI) problems, Epidemiological model problems and some initial value problems (IVPs) of first and second-order ordinary differential equations.

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