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

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

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

If y=e^(m"sin"^(-1)x) , prove that (1-x^(2))(d^(2)y)/(dx^(2))-xdy/(dx)=m^(2)y

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

IF y=e^(tan^(-1)x) then prove that : (1+x^(2))(d^2y)/(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=sin^(-1)x , then show that (1-x^2)\ (d^2y)/(dx^2)-x(dy)/(dx)=0 .

If y=e^m\ sin^((-1)x) , prove that (1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)-m^2y=0 .

If y=Acos(logx)+B sin(logx) then prove that x^(2)(d^(2)y)/(dx^(2))+x(dy)/(dx)+y=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