$(10$ points$)$ A string is fixed at one of its end where the other end is attached to a mass$-$spring system with parameters $m$ and $k$ that is constrained to move only along the vertical direction. If the string has length $l,$ mass per unit length $,$ and is under tension $P,$ derive the characteristic equation from which the natural frequencies of the lateral vibrations can be determined.

Figure $1$: Taught string fixed at one end and attached to a mass$-$spring system at the other end.