Let f be a differentiable function such that x 2 f x - x = 4 ∫ 0 x t f t d t , f 1 = 2 3 . Then 18 f 3 is equal to

Step-by-step solution

Given, x 2 f x - x = 4 ∫ 0 x t   f t   d t Now Differentiating both side w.r.t. x we get, x 2 f ' x + 2 x f x - 1 = 4 x   f x ⇒ x 2 f ' x - 1 = 2 x   f x ⇒ d y d x - 2 x y = 1 x 2 Let   y = f x d y d x = f ' x Which is a linear differential equation, So,   I . F   = e ∫ - 2 x d x = e - 2 ln x = 1 x 2 Now solution of the differential equation is given by, y x 2 = ∫ 1 x 4 d x + C ⇒ y x 2 = - 1 3 x 3 + C Now given, f 1 = 2 3 So, 2 3 = -