Find the equation for the ellipse that satisfies the given conditions:

Ends of major axis $(0, \pm \sqrt{5})$, ends of minor axis $(\pm 1,0)$

Here, the major axis is along 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, $a=\sqrt{5}$ and $b=1$.

Thus, the equation of the ellipse is $\frac{x^{2}}{1^{2}}+\frac{y^{2}}{(\sqrt{5})^{2}}=1$ or $\frac{x^{2}}{1}+\frac{y^{2}}{5}=1$.