A Generalized Series Solution of 𝒏𝒕𝒉 Order Ordinary Differential Equations

Abstract

Differential equations in general play major role in finding solutions to many problems in real life. These real-life problems are modeled by either ordinary differential equations (with uni-variate independent variable) or partial differential equations (with multi- variate independent variables). The solution method adopted is determined by the nature of the differential equation. In this paper, the solution of an 𝑛𝑡ℎ order Ordinary Differential Equation (ODE) is considered. The power series and the conditions for its convergence or otherwise is examined. Also, the index shift in the summation is applied in the simplification of the resulting algebraic expression and with the introduction of the factorial notation, the number of operations required to solve the problem is minimized. The resulting model therefore simplifies the solution method without the rigour of index shit in the summands and algebraic manipulations of the expression obtained. This makes the model applicable to the solution of ordinary differential equation of any order 𝑛. The generalized model is thereafter applied to an ordinary differential equation of order seven without recourse to index shift. This simplified form gives the solution considered and a simple and generalized solution is obtained.

Keywords

• Seriess solution
• Taylor series
• ODE
• Index Shift
• Convergence

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