If x^m + y^m = k, (m 1) and y'' = ax^b y^c such that a + b + c = 0 , then the value of k is...

Step-by-step solution

Differentiating x^m + y^m = k with respect to x : m x^ m-1 + m y^ m-1 y' = 0 y' = -x^ m-1 y^ 1-m Differentiating again with respect to x : y'' = -(m-1) x^ m-2 y^ 1-m - x^ m-1 (1-m) y^ -m y' Substituting y' = -x^ m-1 y^ 1-m : y'' = (1-m) x^ m-2 y^ 1-m - (1-m) x^ m-1 y^ -m (-x^ m-1 y^ 1-m ) y'' = (1-m) x^ m-2 y^ 1-m + (1-m) x^ 2m-2 y^ 1-2m Factoring out (1-m) x^ m-2 y^ 1-2m : y'' = (1-m) x^ m-2 y^ 1-2m (y^m + x^m) Since x^m + y^m = k , we get: y'' = k(1-m) x^ m-2 y^ 2m-1 Comparing this with y'' = ax^b y^c , we obtain