The integrating factor of y + d d x (x y )=x( x+ x) is

Step-by-step solution

Rewriting the given differential equation y + d dx (xy) = x( x + x) in standard form, we apply the product rule to d dx (xy) yielding y + x dy dx . Substituting gives y + y + x dy dx = x( x + x) or 2y + x dy dx = x( x + x) . Dividing by x produces dy dx + 2 x y = x + x , revealing P(x) = 2 x . The integrating factor is e^ P(x) dx = e^ 2 x dx = e^ 2 |x| = x^2 . Final Answer: C