If `y=xlog(x/(a+bx))`, then prove that `x^3(d^2y)/(dx^2)=(x(dy)/(dx)-y)^2`

If y= x log ((x)/(a + bx)) prove that (d^(2)y)/(dx^(2)) = (1)/(x) ((a)/(a+ bx))^(2)

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

If y=Acos(logx)+B sin(logx) then prove that x^(2)(d^(2)y)/(dx^(2))+x(dy)/(dx)+y=0 .

If x= e^((x)/(y)) , then prove that (dy)/(dx)= (x-y)/(x.log x)

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

If y= e^(cos^(-1)x), -1 le x le 1 , then prove that (1-x^(2)) (d^(2)y)/(dx^(2))-x (dy)/(dx)- y= 0

If x = a cos theta + b sin theta and y =asin theta-b cos theta , then prove that y^2 (d^2y)/(dx^2)-x(dy)/(dx)+y=0

Solve y-x(dy)/(dx)=a(y^(2)+(dy)/(dx))

If e^(y) (x+ 1)=1 , show that (d^(2)y)/(dx^(2))= ((dy)/(dx))^(2)