$I=D{\displaystyle \underset{0}{\overset{a}{}}}[r(\frac{d^{2}w}{d{r}^2}{)}^2+\frac{1}{r}(\frac{dw}{dr}{)}^2+2v\frac{dw}{dr}\frac{d^{2}w}{d{r}^2}-\frac{2q}{D}rw]dr$

where $D$ and $v$ are elastic constants, and $wis$ the deflection $(in$ the $z$ direction$)of$ the center$-$

plane $of$ the plate. Show that the following $is$ the proper governing differential equation for $w$ :

$r\frac{d^{4}w}{d{r}^4}+2\frac{d^{3}w}{d{r}^3}-\frac{1}{r}\frac{d^{2}w}{d{r}^2}+\frac{1}{r^2}\frac{dw}{dr}=\frac{qr}{D}$