If f x is a differentiable function in the interval 0 , ∞ such that f 1 = 1 and lim t → x ⁡ t 2 f x - x 2 f t t - x = 1 , for each x > 0 , then f 3 2 is equal to

Step-by-step solution

Let L =   lim t → x ⁡ t 2 f x - x 2 f t t - x = 1 Applying L'Hospital Rule L =   lim t → x ⁡ 2 t   f x - x 2 f ′ t 1 = 1 Or 2 x   f x - x 2 f ′ x = 1  ⇒  d y d x + - 2 x y = - 1 x 2 .......... ( 1 ) Solving the above linear differential equation we get Integrating factor = e ∫ - 2 x d x = e - 2 ln x = 1 x 2 After multiplying the equation 1 by 1 x 2 , and simplifying the equation 1 becomes, d d x y x 2 = - 1 x 4 ⇒ y x 2 = ∫