Answer
$${y^2} - 2y = 2{e^x} - {x^2} + C$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dx}} = \frac{{{e^x} - x}}{{y - 1}} \cr
& {\text{Separating variables leads to}} \cr
& \left( {y - 1} \right)dy = \left( {{e^x} - x} \right)dx \cr
& {\text{To solve this equation}}{\text{, determine the antiderivative of each side}} \cr
& \int {\left( {y - 1} \right)dy} = \int {\left( {{e^x} - x} \right)dx} \cr
& {\text{integrating by using the basic intetgration rules}} \cr
& \frac{{{y^2}}}{2} - y = {e^x} - \frac{{{x^2}}}{2} + k \cr
& {\text{multiply both sides by 2}} \cr
& {y^2} - 2y = 2{e^x} - {x^2} + 2k \cr
& {y^2} - 2y = 2{e^x} - {x^2} + C \cr} $$
