using Rayleigh$-$Ritz, with the provided displacement equation w$($x$)=$ C$1($x$3$ Lx$2)+$C$2($x$4$ L$2$x$2)$ determine the following:

$1.$ Pcr

$2.$ The mode shape

Note that the spring is located at x $=$ L$/2$

L $=1$ m

E $=200$GPa

I $=1\mathrm{.}257\backslash$times $107$ m$4$

A $=0\mathrm{.}0004$ m$2$

q$0=100$ N

Kf $=1000$ N$/$m