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Highlights• The paper focused on the numerical solution of higher order initial value problems.• Power series was used as the basis function for the derivation of the method.• The method solved second, third and fourth order ordinary differential equations concurrently.• This method satisfied the basic properties of a linear multistep method.• This method generated more accurate results than the existing numerical methods.

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The development and application of an implicit hybrid block method for the direct solution of second order ordinary differential equations with given initial conditions is shown in this research. The derivation of the three-step scheme was done through collocation and interpolation of power series approximation to give a continuous linear multistep method. The evaluation of the continuous method at the grid and off grid points formed the discrete block method. The basic properties of the method such as order, error constant, zero stability, consistency and convergence were properly examined. The new block method produced more accurate results when compared with similar works carried out by existing authors on the solution of linear and non-linear second order ordinary differential equations

This work considers the direct solution of general third order ordinary differential equation by three-step irrational linear multistep method. This method is derived using collocation and interpolation techniques. An irrational three-step method is developed. Taylor series and block methods are used to generate the independent solution at selected points. The properties of the method were also determined. The developed method was applied on general third order ordinary differential equations. And the performance of the numerical results of the method compared favourably with the results of existing authors in the recent literature to test its accuracy and stability.