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