Find the equation of the hyperbola satisfying the give conditions:

Vertices $(0, \pm 3)$, foci $(0, \pm 5)$

Here, the vertices are on the $y$-axis.

Therefore, the equation of the hyperbola is of the form $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$.

Since the vertices are $(0, \pm 3), a=3$.

Since the foci are $(0, \pm 5), c=5$.

We know that $a^{2}+b^{2}=c^{2}$.

$\therefore 3^{2}+b^{2}=5^{2}$

$\Rightarrow b^{2}=25-9=16$

Thus, the equation of the hyperbola is $\frac{y^{2}}{9}-\frac{x^{2}}{16}=1$.