If `y = a sin (log x) + b cos (log x)`, then prove that : `x^2 (d^2y)/dx^2+ x dy/dx+ y = 0`.

If y= a cos (log x)+b sin (log x) , show that x^(2) (d^(2)y)/(dx^2)+x(dy)/(dx)+y=0 .

If y = sin(2 cos^(-1)x) , then prove that (1 - x^2)(d^2y)/(dx) -x dy/dx + 4y = 0

Find the second order derivative of the following functions If y = a cos (log x) + b sin (log x), prove that x^2y_2 + xy_1 + y = 0 .

If y= (cos^-1 x)^2 , prove that: (1 - x^2 ) ((d^2y)/dx^2) - x(dy/dx) -2 = 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=log (x+ sqrt ( x^2+1)), prove that: (x^2+1) d^2y/dx^2 + xdy/dx=0 .

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

If y= sin (2 sin^-1 x) , prove that: (1 - x^2 ) (d^2y/dx^2) - x(dy/dx) + 4y = 0 .