Elliptic integrals solution to elastica's boundary value problem of a rod bent by axial compression

In this article we solve in closed form a nonlinear differential equation modelling the planar, non-inflectional elastica (deflection $y$) of a thin, flexible, simply supported, $x$-straight rod, which withdraws bending and doesn't retain its original length $L$ when deformed under a compressive thrust. We solve a Dirichlet two-point boundary value problem providing an explicit inverse function $x(y)$ through elliptic integrals of I and II kind. Integration is performed twice via the ``shooting'' getting both branches of elastica. In such a way $y(x)$ is required to be invertible, and then our solution is found within each of two monotonicity $x$-ranges for $y$% , before and beyond the bifurcation value $x^{*}=L/2$. An auxiliary unknown is then introduced and successively computed through a suitable welding condition. In such a way $x=x(y)$ is completely known in its two symmetrical branches.