If y + d d x (x y )=x( x+ x) then

Step-by-step solution

The differential equation y + d dx (xy) = x( x + x) simplifies using the product rule: d dx (xy) = y + x dy dx . Substituting gives y + y + x dy dx = x( x + x) , which rearranges to x dy dx + 2y = x( x + x) . Dividing by x yields the standard linear form: dy dx + 2 x y = x + x . The integrating factor is e^ 2 x dx = x^2 . Multiplying through and integrating gives y x^2 = x^2( x + x) dx + C . Evaluating the integral by parts: x^2 x dx = -x^2 x + 2x x + 2 x and x^2 x dx = x^3 3 x - x^3 9 . Combining results: y x^2 =