Chapter 6.2, Problem 35E

Vibrating Beam. In studying the transverse vibrations of a beam, one encounters the homogeneous equation

$EI\frac{d^{4}y}{d{x}^4}-ky=0$,

where $y\left(x\right)$ is related to the displacement of the beam at position $x$, the constant $E$ is Young’s modulus, $I$ is the area moment of inertia, and $k$ is a parameter. Assuming $E$, $I$, and $k$ are positive constants, find the general solution in terms of sines, cosines, hyperbolic sines, and hyperbolic cosines.