A uniform cable hangs from the ceiling with the lower end loose. Derive the eigenvalue problem for the lateral vibration of the cable and obtain a closed$-$form solution of the problem. Calculate the first three natural frequencies and plot the first three natural modes. Solve by the Rayleigh$-$Ritz method by assuming the trial functions Y$($x$)=[1-($x$/$L$)^2],$ Y$($x$)=$a$[1-($x$/$L$)^2]+$ a$2[1-($x$/$L$)^3]$ and Y$($x$)=$ a$[1-($x$/$L$)2]+$ a$2[1-($x$/$L$)^3]+$ a$3[1-($x$/$L$)],$ in sequence. And compare the answers.