If `ysqrt(x^2 +1) = log{sqrt(x^2 +1)- x}` then prove that `(x^2 +1) dy/dx + xy + 1 = 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 .

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 = tan^(-1) x, prove that (1 + x^2)y_2 + 2xy_1 = 0

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

Find dy/dx xy^2 + x^2y + 1 = 0

Solve : dy/dx (x^2 - 1) + 2xy = 1 .

If y =(sin^(-1) x)^2 , prove that (1-x^2)y_2 - xy_1 - 2 = 0

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

Solve: (x^2-1)dy/dx+2xy=1