Answer
$$
\left(\frac{x}{5}\right)^{2}+\left(\frac{y}{7}\right)^{2}=1.
$$
Work Step by Step
Since the vertices are at $(\pm 5,0)$ and $(0,\pm7)$, then we have
$$a=5, \quad b=7$$
and hence the ellipse is centered at the origin and has the equation
$$
\left(\frac{x}{5}\right)^{2}+\left(\frac{y}{7}\right)^{2}=1.
$$
