Let y = y x be the solution curve of the differential equation d y d x + 1 x 2 - 1 y = x - 1 x + 1 1 2 , x > 1 passing through the point 2 , 1 3 . Then 7 y 8 is equal to

Step-by-step solution

Given, d y d x + y x 2 - 1 = x - 1 x + 1 1 2 is a linear differential equation. Here I . F . = e ∫ d x x 2 - 1 = e 1 2 ln x - 1 x + 1 = x - 1 x + 1 General solution will be y x - 1 x + 1 = ∫ x - 1 x + 1 d x + C ⇒ y x - 1 x + 1 = x - 2 ln x + 1 + c Given y 2 = 1 3 ⇒   c = 2 ln 3 - 5 3 So, the equation of the curve is y x - 1 x + 1 = x - 2 ln x + 1 + 2 ln 3 - 5 3 Now putting x = 8 , we get 7 y 8 3 = 8 - 4 ln 3 + 2 ln 3 - 5 3 ⇒   7 y 8 = 19 - 6 ln 3