" 9.If "y=log[x+sqrt(x^(2)+1)]" ,then prove that ":(x^(2)+1)(d^(2)y)/(dx^(2))+x(dy)/(dx)=0

If y=log[x+sqrt(x^(2)+1)], prove that (x^(2)+1)(d^(2)y)/(dx^(2))+x(dy)/(dx)=0

(i) If y^(1//m) + y^(-1//m) = 2x then prove that (x^(2)-1) (d^(2)y)/(dx^(2)) + x (dy)/(dx) - m^(2)y =0 (ii) If y = ln (x + sqrt(1+x^(2)) , then prove that (1+x^(2)) (d^(2)y)/(dx^(2)) + x (dy)/(dx)=0

(i) If y^(1//m) + y^(-1//m) = 2x , then prove that (x^(2)-1) (d^(2)y)/(dx^(2)) + x (dy)/(dx) - m^(2)y =0 (ii) If y = ln (x + sqrt(1+x^(2))) , then prove that (1+x^(2)) (d^(2)y)/(dx^(2)) + x (dy)/(dx)=0

If y=log (x + sqrt(x^(2) + 1)) then show that, (x^(2) + 1) (d^(2)y)/(dx^(2)) + x (dy)/(dx)= 0

If y=sin^(-1)x ,then prove that (1-x^(2))(d^(2)y)/(dx^(2))=x(dy/dx)

If y=log{x+sqrt(x^(2)+a^(2))}, prove that: (x^(2)+a^(2))(d^(2)y)/(dx^(2))+x(dy)/(dx)=0

If y=log(x+sqrt(1+x^(2))), then show that (1+x^(2))(d^(2)y)/(dx^(2))+x(dy)/(dx)=0

If y=sin^(-1)x then prove that (1-x^(2))(d^(y))/(dx^(2))-x(dy)/(dx)=0

If y=sqrt(x+1)-sqrt(x-1) , then prove that (x^(2)-1)(d^(2)y)/(dx^(2))+x(dy)/(dx)-1/4y=0 .