The integrating factor of the differential equation x dy d x + y x=x e ^x x^ -1 2 x(x>0) is

Step-by-step solution

The differential equation x d y d x + y x = x e^ x x^ -1/2 x is rearranged into standard linear form by dividing through by x , yielding d y d x + x x y = e^ x x^ -3/2 x From this, P(x) = x x . The integrating factor is e^ P(x) , d x = e^ x x , d x . Using the substitution u = x , d u = 1 x , d x , the integral evaluates to ( x)² 2 , so the integrating factor simplifies to e^ ( x)²/2 . Analyzing the options, x^ x = e^ ( x)² and ( x )^ x = e^ 1 2 ( x)² ; only C gives the correct form. The integrating factor correspo