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`

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

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

If x=2costheta-cos2\ theta and y=2sintheta-sin2\ theta , prove that (dy)/(dx)=tan((3theta)/2)

If y= A sin x + B cos x , then prove that (d^(2)y)/(dx^(2)) +y= 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= 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 sin^(3)theta + y cos^(3)theta = sintheta costheta and x sin theta = y cos theta , prove that x^(2) + y^(2) = 1.

Find (dy)/(dx) : x=2 cos theta- cos^(2) theta and y=2 sin theta- sin 2theta Show that (dy)/(dx)= -1 " when " theta = (pi)/(2)