Answer
$$y = \frac{1}{2}{e^{\frac{{{x^2}}}{2}}}$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dx}} = xy,{\text{ }}\left( {0,\frac{1}{2}} \right) \cr
& {\text{Separate the variables}} \cr
& \frac{{dy}}{y} = xdx \cr
& {\text{Integrate both sides}} \cr
& \int {\frac{1}{y}dy} = \int x dx \cr
& \ln \left| y \right| = \frac{{{x^2}}}{2} + C{\text{ }}\left( {\bf{1}} \right) \cr
& {\text{Use the initial condition }}\left( {0,\frac{1}{2}} \right) \cr
& \ln \left| {\frac{1}{2}} \right| = \frac{{{{\left( 0 \right)}^2}}}{2} + C \cr
& C = \ln \left( {\frac{1}{2}} \right) \cr
& {\text{Substitute }}C{\text{ into }}\left( {\bf{1}} \right) \cr
& \ln \left| y \right| = \frac{{{x^2}}}{2} + \ln \left( {\frac{1}{2}} \right) \cr
& {\text{Solve for }}y \cr
& y = \frac{1}{2}{e^{\frac{{{x^2}}}{2}}} \cr
& \cr
& {\text{Graph}} \cr} $$
