Answer
$$y = \frac{1}{6}{x^3} - {x^2} + 2$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dx}} = \frac{1}{2}{x^2} - 2x \cr
& {\text{Separate the variables}} \cr
& dy = \left( {\frac{1}{2}{x^2} - 2x} \right)dx \cr
& {\text{Integrate both sides}} \cr
& \int {dy} = \int {\left( {\frac{1}{2}{x^2} - 2x} \right)} dx \cr
& y = \frac{1}{6}{x^3} - {x^2} + C{\text{ }}\left( {\bf{1}} \right) \cr
& {\text{Use the initial condition }}\left( {6,2} \right) \cr
& 2 = \frac{1}{6}{\left( 6 \right)^3} - {\left( 6 \right)^2} + C \cr
& 6 = 36 - 36 + C \cr
& C = 2 \cr
& {\text{Substitute }}C{\text{ into }}\left( {\bf{1}} \right) \cr
& y = \frac{1}{6}{x^3} - {x^2} + 2 \cr
& \cr
& {\text{Graph}} \cr} $$
