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This paper shows how to obtain highly accurate solutions of eighth-orderboundary-value problems of linear and nonlinear ordinary differentialequations. The presented method is based on the Theory of FunctionalConnections, and is solved in two steps. First, the Theory of FunctionalConnections analytically embeds the differential equation constraints into acandidate function (called a $constrained \, expression$) that contains afunction that the user is free to choose. This expression always satisfies theconstraints, no matter what the free function is. Second, the free-function isexpanded as a linear combination of orthogonal basis functions with unknowncoefficients. The constrained expression (and its derivatives) are thensubstituted into the eighth-order differential equation, transforming theproblem into an unconstrained optimization problem where the coefficients inthe linear combination of orthogonal basis functions are the optimizationparameters. These parameters are then found by linear/nonlinear least-squares.The solution obtained from this method is a highly accurate analyticalapproximation of the true solution. Comparisons with alternative methodsappearing in literature validate the proposed approach.

Least-squares Solutions of Eighth-order Boundary Value Problems using the Theory of Functional Connections