If x = (t), y = (pt) , then (1 - x^2) d^2 y dx^2 - x dy dx =

Step-by-step solution

Given x = (t) and y = (pt) t = ⁻¹(x) Substituting t in y : y = (p ⁻¹(x)) Differentiating with respect to x : dy dx = - (p ⁻¹(x)) p 1 - x^2 1 - x^2 dy dx = -p (p ⁻¹(x)) Differentiating again with respect to x : 1 - x^2 d^2 y dx^2 + dy dx ( -x 1 - x^2 ) = -p (p ⁻¹(x)) p 1 - x^2 Multiplying the entire equation by 1 - x^2 : (1 - x^2) d^2 y dx^2 - x dy dx = -p^2 (p ⁻¹(x)) Since y = (p ⁻¹(x)) , we get: (1 - x^2) d^2 y dx^2 - x dy dx = -p^2 y Answer: -p^2 y