If `y = (sqrt(x+1)+sqrt(x-1))^(1/2)`, then prove that `(x^2-1)d/dx(dy/dx)+x(dy)/(dx)=(1/16)y`

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

Differentiate the following w.r.t.x. y = x^(x^(x^( oo))) , prove that x (dy)/(dx) = (y^2)/(1 - y log x)

Solve the differential equation (x-1)dy+ydx=x(x-1)y^(1//3)dx .

Differentiate the following w.r.t.x. y = e^(x^(+ex + ex^(-"to" oo))), prove that (dy)/(dx) = y/(1 - y)

Differentiate the following w.r.t.x. If x^y = y^x , prove that (dy)/(dx) = (y/x - "log y")/(x/y- "log x")

Solve the differential equations: x^2(dy)/(dx)=y^2+2xy,y(1)=1

Solve the differentiate equation (dy)/(dx)=1/(1+x^(2)),y(0)=3 .

Differentiate the following w.r.t.x. If y^x = e^(y - x) , prove that (dy)/(dx) = ((1 + log y)^2)/(log y)