Answer
$$y = 3{\sin ^{ - 1}}t - \pi \,\,$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dt}} = \frac{3}{{\sqrt {1 - {t^2}} }},\,\,\,\,y\left( {\frac{{\sqrt 3 }}{2}} \right) = 0 \cr
& {\text{Separating variables}} \cr
& dy = \frac{3}{{\sqrt {1 - {t^2}} }}dt \cr
& {\text{Integrating both sides of the equation}} \cr
& \int {dy} = \int {\frac{3}{{\sqrt {1 - {t^2}} }}} dt \cr
& y = 3\int {\frac{1}{{\sqrt {1 - {t^2}} }}} dt \cr
& y = 3{\sin ^{ - 1}}t + C\,\, \cr
& \cr
& {\text{Using the initial condition }}y\left( {\frac{{\sqrt 3 }}{2}} \right) = 0 \cr
& 0 = 3{\sin ^{ - 1}}\left( {\frac{{\sqrt 3 }}{2}} \right) + C \cr
& 0 = 3\left( {\frac{\pi }{3}} \right) + C \cr
& C = - \pi \cr
& \cr
& {\text{Then}}{\text{,}} \cr
& y = 3{\sin ^{ - 1}}t - \pi \,\, \cr} $$
