If x= t and y = pt , then the value of (1-x^2 ) d^2 y d x^2 -x d y d x + p ^2 y =

Step-by-step solution

Given x = t and y = pt , evaluate (1-x^2 ) d^2 y d x^2 -x d y d x +p^2 y . Since x = t , we have t = x for t [- /2, /2] , and y = (p x) . The first derivative is dy dx = p (p x) 1-x^2 . Multiplying both sides by 1-x^2 gives 1-x^2 dy dx = p (p x) . Squaring both sides yields (1-x^2) ( dy dx )^2 = p^2 ^2(p x) . Using ^2 = 1 - ^2 , this becomes (1-x^2) ( dy dx )^2 = p^2 (1 - y^2) . Differentiating both sides with respect to x gives: -2x ( dy dx )^2 + 2(1-x^2) dy dx d^2y dx^2 = -2p^2 y dy dx . Dividing through by 2 dy