See the fixed$-$fixed beam illustrated below. No forces are applied.

a$)$ Solve for the beam's exact natural frequencies.

b$)$ Solve for the beam's natural frequencies using a virtual displacement $($two modes$).$

c$)$ Solve for the beam's natural frequencies using the finite element method $($two elements$).$

Provide a thorough step$-$by$-$step workthrough analytically and solve using MATLAB$/$python to compare the results