Chapter 1, Problem 1.12P

A differential equation encountered in the vibration of beams is

$\frac{d^{4}y}{d{x}^4}=\frac{{\omega }^2\gamma }{D}\gamma$ where $x$ = distance measured along the beam; $\left[x\right]=\left[L\right]$ $y$ = displacement of the beam; $\left[y\right]=\left[L\right]$ $\omega$ = circular frequency of vibration; $\left[\omega \right]=\left[T^{-1}\right]$ $\gamma$ = mass of the beam per unit length; $\left[\gamma \right]=\left[M{L}^{-1}\right]$ $D$ = bending rigidity of beam; $\left[D\right]=\left[F{L}^2\right]$ Show that the equation is dimensionally homogeneous. Note that $\left[F\right]=\left[ML{T}^{-2}\right]$ see Eq. (1.21:).