`y=sqrt((1-x)/(1+x))`prove that `(1-x^(2))(dy)/(dx)+y=0`

Given that y= sqrt( (1-x)/( 1+x)) show that (1-x^(2) ) (dy)/( dx) + y=0

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

If y= y=sqrt((1-sin 2x )/(1+sin 2 x)) prove that (dy)/(dx)+sec^2(pi/4-x)=0.

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

If y = sqrt(1+sqrt(1+x^(4))) , prove that y * (y^(2) - 1)(dy)/(dx) = x^(3) .

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

if y=sqrt((1-cos x )/(2)) , the n prove that (dy)/(dx)=1/2cos" x/2.

(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)) , prove that 2x(dy)/(dx)=sqrt(x)-1/(sqrt(x))