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

Step-by-step solution

y=x ( 1 a x + 1 a ) Then d y d x =x 1 ( 1 a x + 1 a ) (- a a x^2 )+ ( 1 a x + 1 a ) aligned & d y d x = -a x a x(1+x) + ( 1 a x + 1 a ) & d y d x =- 1 (1+x) + ( 1 a x + 1 a ) aligned and d^2 y d x^2 =+ 1 (x+1)^2 + 1 ( 1 a x + 1 a ) (- 1 a x^2 ) aligned & = 1 (x+1)^2 - a x (1+x) (a x^2 ) & d y d x^2 = 1 (x 1)^2 - 1 x(x 1) aligned Now, x(x+1) d^2 y d x^2 +x d y d x -y=x(x+1) [ 1 (x+1)^2 - 1 x(x+1) ]+x [- 1 (1+x) + ( 1 a x + 1 a ) ]-x ( 1 a x + 1 a ) aligned & = x x+1 -1- x x+1 +x ( 1 a x + 1 a )-x ( 1 a x + 1 a ) & =