If `x= cot sqrty`, prove that `(x^(2) +1)^(2) (d^(2)y)/(dx^(2)) + 2x (x^(2) + 1) (dy)/(dx)-2 = 0`

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

If y=(cot^(-1)x)^(2) , show that (1+x^(2))^(2).(d^(2)y)/(dx^(2))+2x(1+x^(2)).(dy)/(dx)=2

If y= cot x, prove that (d^(2)y)/(dx^(2)) + 2y (dy)/(dx)= 0.

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

If y=e^("mcos"^(-1)x) , prove that: (1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)=m^(2)y

If x=tan(1/a log y) , prove that (1+x^2) (d^2y)/(dx^2)+(2x-a) (dy)/(dx)=0

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

If x y=1 , prove that (dy)/(dx)+y^2=0 .