Answer
$\dfrac{y^2}{4}-\dfrac{x^2}{9}=1$
Work Step by Step
Center: $(0,0)$ and vertices: $(0,-2); (0,2)$
This yields: $ a=2$ and Asymptote: $ y=\dfrac{2}{3} x \implies b=2$
And we have a horizontal traverse axis.
Standard Equation for a hyperbola is: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$
We have $ a^2=9; b^2 =4$
So, the equation for the hyperbola is:
$\dfrac{y^2}{4}-\dfrac{x^2}{9}=1$
