Let y=y(x) be a differentiable function in the interval (0, ) such that y(1)=2 , and _ t x ( t² y(x)-x² y(t) x-t )=3 for each x>0 . Then 2 y(2) is equal to

Step-by-step solution

Given the limit _ t x t^2 y(x) - x^2 y(t) x - t = 3 . The limit is in 0 0 form. Applying L'Hopital's Rule with respect to t : _ t x d dt (t^2 y(x) - x^2 y(t)) d dt (x - t) = 3 _ t x 2t y(x) - x^2 y'(t) -1 = 3 Substituting t = x : 2x y(x) - x^2 y'(x) -1 = 3 x^2 y'(x) - 2x y(x) = 3 Dividing by x^4 to make it a linear differential equation or recognizing the quotient rule form: x^2 y'(x) - 2x y(x) x^4 = 3 x^4 d dx ( y(x) x^2 ) = 3 x^4 Integrating both sides with respect to x : y(x) x^2 = 3x⁻⁴ dx = 3x⁻³ -3 + C = - 1 x^