Find the equation for the ellipse that satisfies the given conditions: Vertices (0, ±13), foci (0, ±5)

Vertices $(0, \pm 13)$, foci $(0, \pm 5)$

Here, the vertices are on the y-axis.

Therefore, the equation of the ellipse will be of the form $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$, where $a$ is the semi-major axis.

Accordingly, a = 13 and c = 5.

It is known that $a^{2}=b^{2}+c^{2}$.

$\therefore 13^{2}=b^{2}+5^{2}$

$\Rightarrow 169=b^{2}+25$

$\Rightarrow b^{2}=169-25$

$\Rightarrow b=\sqrt{144}=12$

Thus, the equation of the ellipse is $\frac{x^{2}}{12^{2}}+\frac{y^{2}}{13^{2}}=1$ or $\frac{x^{2}}{144}+\frac{y^{2}}{169}=1$.