If `y =(ax +b)/( cx + d)` , then prove that `2y_1y_3=3(y_2)^2`

If f(x)= y = (ax-b)/(cx-a) , then prove that f(y) = x

If y= A sin x + B cos x , then prove that (d^(2)y)/(dx^(2)) +y= 0

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

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

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= e^(ax) sin bx then show that, y_(2)- 2ay_(1) + (a^(2) + b^(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 .

y=[log(x+sqrt(x^2+1))]^2 then prove that (x^2+1)y_2+x y_1=2