Let f : [ 1 . ∞ ) → ℝ be a differentiable function such that f 1 = 1 3 and 3 ∫ 1 x f t d t = x f x - x 3 3 , x ∈ [ 1 , ∞ ) . Let e denote the base of the natural logarithm. Then the value of f e is

Step-by-step solution

Given, 3 ∫ 1 x f t d t = x f x - x 3 3 Now differentiating both side we get, ⇒ 3 f x = f x + x f ' x − x 2 ⇒ x f ' x − 2 f x = x 2 ⇒ f ' x − 2 x f x = x Which is a linear differential equation, ⇒ I . F . = e − 2 x d x = 1 x 2 Now solution is given by, ⇒ y 1 x 2 = ∫ x × 1 x 2 d x = ln ⁡ x + C ⇒ y = x 2 ( ln ⁡ x + C ) ⇒ f x = x 2 ( ln x + C ) Now using given value, f 1 = 1 3 we get, ⇒ 0 + C = 1 3 ⇒ C