If x= e ^ ⁻¹ t and y= e ^ ⁻¹ t , then d ^2 y ~d x^2 is

Step-by-step solution

aligned x y & = e ^ ⁻¹ t e ^ ⁻¹ t & = e ^ ⁻¹ t + ⁻¹ t x y & = e ^ 2 aligned Differentiating both sides w.r.t. x , we get aligned & x d y ~d x +y 1=0 & d y ~d x =- y x ...(i) aligned aligned d ^2 y ~d x^2 & =- ( x d y ~d x -y 1 x^2 ) d ^2 y ~d ^2 & =- ( x (- y x )-y x^2 )... [From (i)] & = 2 y x^2 aligned