Let y=y(x) be the solution of the differential equation x ~d y ~d x -y=x² x, x (0, ) . If y ( 2 )= 2 , then 6 y ( 6 )-8 y ( 4 ) is equal to :

Step-by-step solution

The differential equation x dy dx - y = x^2 x can be rewritten as dy dx - y x = x x . The integrating factor is (x) = 1 x . Multiplying by (x) gives d dx ( y x ) = x . Integrating: y x = | x| + C , so y = x ( x) + Cx . Using y ( 2 ) = 2 : we get 2 = 0 + C 2 , thus C = 1 . Therefore y = x(1 + ( x)) . Now: y ( 6 ) = 6 (1 - 2) and y ( 4 ) = 4 (1 - 2 2 ) . Thus 6y ( 6 ) - 8y ( 4 ) = (1 - 2) - 2 (1 - 2 2 ) = - .