An elastic bar of length $L$ spins with angular velocity $$ about an axis, as shown in the figure below. The radial acceleration at a generic point $x$ along the bar is $a(x)={}^{2}x.$ Due to this radial acceleration, the bar stretches along $x$ with displacement function $u(x).$ The displacement $u(x)$ is governed by the following equations:

$\frac{d}{dx}((x))+a(x)=0$ PDE

$(x)=E\frac{du}{dx}$ Hooke's law

where $(x)$ is the axial stress in the rod, $$ is the mass density, and $E$ is the $($constant$)$ Young's modulus. The bar is PINNED on the rotation axis at $x=0,$ and IT IS FREE AT $x=L.$