Answer
$$y = - \frac{1}{3}{\left( {9 - {x^2}} \right)^{3/2}} + 5$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dx}} = x\sqrt {9 - {x^2}} \cr
& {\text{Separate the variables}} \cr
& dy = x\sqrt {9 - {x^2}} dx \cr
& {\text{Integrate both sides}} \cr
& \int {dy} = \int {x\sqrt {9 - {x^2}} } dx \cr
& y = - \frac{1}{2}\int {\left( { - 2x} \right)\sqrt {9 - {x^2}} } dx \cr
& y = - \frac{1}{2}\left[ {\frac{{{{\left( {9 - {x^2}} \right)}^{3/2}}}}{{3/2}}} \right] + C \cr
& y = - \frac{1}{3}{\left( {9 - {x^2}} \right)^{3/2}} + C{\text{ }}\left( {\bf{1}} \right) \cr
& {\text{Use the initial condition }}\left( {0, - 4} \right) \cr
& - 4 = - \frac{1}{3}{\left( {9 - {0^2}} \right)^{3/2}} + C \cr
& - 4 = - 9 + C \cr
& C = 5 \cr
& {\text{Substitute }}C{\text{ into }}\left( {\bf{1}} \right) \cr
& y = - \frac{1}{3}{\left( {9 - {x^2}} \right)^{3/2}} + 5 \cr
& \cr
& {\text{Graph}} \cr} $$
