Answer
$$ y = \frac{3}{2}{\tan ^{ - 1}}\left( {\frac{x}{2}} \right) - \frac{3}{8}\pi $$
Work Step by Step
$$\eqalign{
& \left( {{x^2} + 4} \right)\frac{{dy}}{{dx}} = 3,\,\,\,\,y\left( 2 \right) = 0 \cr
& {\text{Separate the variables}} \cr
& \frac{{dy}}{{dx}} = \frac{3}{{{x^2} + 4}} \cr
& {\text{integrate both sides}} \cr
& y = \int {\frac{3}{{{x^2} + 4}}} dx \cr
& y = \frac{3}{2}{\tan ^{ - 1}}\left( {\frac{x}{2}} \right) + C\,\,\,\left( {\bf{1}} \right) \cr
& \cr
& {\text{use initial condition }}y\left( 2 \right) = 0 \cr
& 0 = \frac{3}{2}{\tan ^{ - 1}}\left( {\frac{2}{2}} \right) + C \cr
& 0 = \frac{3}{2}\left( {\frac{\pi }{4}} \right) + C \cr
& C = - \frac{3}{8}\pi \cr
& \cr
& {\text{Then}}{\text{, substituting }}C = - \frac{3}{8}\pi {\text{ in }}\left( {\bf{1}} \right) \cr
& y = \frac{3}{2}{\tan ^{ - 1}}\left( {\frac{x}{2}} \right) - \frac{3}{8}\pi \cr} $$
