If y= ( ⁻¹ x )^2+ ( ⁻¹ x )^2 , then (1-x^2 ) d^2 y d x^2 -x d y d x =

Step-by-step solution

Given y = ( ⁻¹ x)^2 + ( ⁻¹ x)^2 . Using the identity ⁻¹ x = 2 - ⁻¹ x , we have y = ( ⁻¹ x)^2 + ( 2 - ⁻¹ x)^2 . Let u = ⁻¹ x . Then y = u^2 + ( 2 - u)^2 = u^2 + ^2 4 - u + u^2 = 2u^2 - u + ^2 4 . Differentiating with respect to x : dy dx = dy du du dx = (4u - ) 1 1-x^2 . Thus, 1-x^2 dy dx = 4 ⁻¹ x - . Differentiating both sides with respect to x again: 1-x^2 d^2y dx^2 + dy dx -2x 2 1-x^2 = 4 1 1-x^2 . Multiplying the entire equation by 1-x^2 : (1-x^2) d^2y dx^2 - x dy dx = 4 . Answer: 4