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

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

If y= tan^-1 x then prove that (1+x^2) y_(2) + 2xy_(1) =0. Differentiate sec^-1(1/(2x^2 -1)) with respect to sqrt 1-x^2 .

If y = sin(mcos^(-1)x) then prove that (1-x^(2))y_(2)-xy_(1)+m^(2)y=0 .

If y = ax sin x show that x^2y_2 - 2xy_1 + (x^2 + 2)y = 0

If y^(1//m)+y^(-1//m)=2x , then prove that (x^(2)-1)y_(2)+xy_(1)=m^(2)y^(2)

If 2x=y^(1/m)+y^(-1/m) , then prove that (x^2-1)y_(n+2)+(2n+1)xy_(n+1)+(n^2-m^2)y_n=0 .

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

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

If ysqrt(x^2 +1) = log{sqrt(x^2 +1)- x} then prove that (x^2 +1) dy/dx + xy + 1 = 0

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