Answer
$$\frac{{{y^2}}}{{36}} - \frac{{{x^2}}}{{144}} = 1$$
Work Step by Step
$$\eqalign{
& {\text{vertices at }}\left( {0,6} \right),\left( {0, - 6} \right);{\text{ asymptotes }}y = \pm \frac{1}{2}x \cr
& {\text{The }}x{\text{ - coordinate in the vertices are the same, then the hyperbola}} \cr
& {\text{has the equation }}\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1 \cr
& {\text{With vertices at }}\left( {0, \pm a} \right){\text{ and asymptotes }}y = \pm \frac{a}{b}x \cr
& {\text{,Then}} \cr
& {\text{vertices }}\left( {0, - 6} \right),\left( {0,6} \right) \to \,\,\,a = 6 \cr
& y = \pm \frac{a}{b}x = \pm \frac{1}{2}x \cr
& \frac{a}{b} = \frac{1}{2} \cr
& \frac{6}{b} = \frac{1}{2} \cr
& b = 12 \cr
& {b^2} = 144 \cr
& {\text{The equation of the hyperbola is}} \cr
& \frac{{{y^2}}}{{36}} - \frac{{{x^2}}}{{144}} = 1 \cr} $$
