Solving nth Order Differential Equations and Polynomial Equations of nth Degree by Using Unified Formulas Composed of Radical Expressions

This paper presents new theorems solving differential equations of nth-order, where the possibility of calculating solutions nearly in parallel is considered. These theorems are based on an engineering methodology that forwards the concept of solutions architecting according to an engineering approach, where the process of developing expressions and sub expressions of solutions is based on requirements engineering, analysis, design, and then developing the complete algebraic formulas of solutions to be scalable and projectable on any orders or degrees of equations. The new engineering methodology in this paper is initially developed to solve nth degree polynomial equations in general forms while using specific new unified formulas composed of radical expressions, which allow calculating the roots nearly in parallel. Then, this paper forwards this engineering methodology by using the roots of nth degree polynomial equations in general forms to solve differential equations of nth order. This methodology presents specific logic, statements, conditions, mathematical expressions, and new unified formulas that allow calculating the solutions of nth degree polynomials and nth-order differential equations. In addition, this paper presents the results of applying this engineered methodology to differential and polynomial equations of fourth degree, fifth degree, and sixth degree. This methodology is built on precise details that provide step-by-step logic and formulas to calculate the solutions, which allow concretizing multiple theorems, formulating the algebraic expressions of all solutions for different orders and degrees of equations, where the possibility of calculating the values of these solutions nearly in parallel while relying on distributed structures of terms.