Let y = y(x) be the solution of the differential equation x dy dx - y = -2x^3 _e x for x > 0 , satisfying y(1) = 1 2 . The maximum value of y(x) for x > 0 is

Step-by-step solution

Given differential equation: x dy dx - y = -2x^3 _e x Dividing by x , we get the standard linear differential equation form: dy dx - 1 x y = -2x^2 _e x The integrating factor (I.F.) is: I.F. = e^ - 1 x , dx = e^ - _e x = 1 x The solution is given by: y 1 x = (-2x^2 _e x 1 x ) dx y x = -2 x _e x , dx Using integration by parts: y x = -2 ( _e x x^2 2 - 1 x x^2 2 , dx ) y x = -x^2 _e x + x^2 2 + C Given y(1) = 1 2 , we substitute x = 1 and y = 1 2 : 1/2 1 = -1^2 _e 1 + 1^2 2 + C 1 2 = 0 + 1 2 + C C = 0 Thus, the parti