  This paper shows how to obtain highly accurate solutions of eighth-order
boundary-value problems of linear and nonlinear ordinary differential
equations. The presented method is based on the Theory of Functional
Connections, and is solved in two steps. First, the Theory of Functional
Connections analytically embeds the differential equation constraints into a
candidate function (called a $constrained \, expression$) that contains a
function that the user is free to choose. This expression always satisfies the
constraints, no matter what the free function is. Second, the free-function is
expanded as a linear combination of orthogonal basis functions with unknown
coefficients. The constrained expression (and its derivatives) are then
substituted into the eighth-order differential equation, transforming the
problem into an unconstrained optimization problem where the coefficients in
the linear combination of orthogonal basis functions are the optimization
parameters. These parameters are then found by linear/nonlinear least-squares.
The solution obtained from this method is a highly accurate analytical
approximation of the true solution. Comparisons with alternative methods
appearing in literature validate the proposed approach.
