If `x=sintheta` , `y=cosptheta` , prove that `(1-x^2)y_2-x y_1+p^2y=0` , where `y_2=(d^2y)/(dx^2)` and `y_1=(dy)/(dx)` .

If x=sin theta,y=cos p theta ,prove that (1-x^(2))y_(2)-xy_(1)+p^(2)y=0, where y_(2)=(d^(2)y)/(dx^(2)) and y_(1)=(dy)/(dx)

If x=sintheta" and "y=cosptheta , p is constant, prove that (1-x^(2))y_(2)-xy_(1)+p^(2)y=0.

If x=sint ,y=sinpt , prove that (1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+p^2y=0.

If x=sint,y=sinpt , prove that (1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+p^2 y=0 .

If y=x^(n-1) log x , prove that (x^2y_2)+(3-2n)xy_1+(n-1)^2 .y=0 where y_1=dy/dx and y_2=(d^2)/(dx^2) .

If x=a(theta+sintheta) , y=a(1+costheta) , prove that (d^2y)/(dx^2)=-a/(y^2) .

If x=a(theta+sintheta),y=a(1+costheta), Prove that (d^2y)/(dx^2)=-a/(y^2)

If y=A sin^2x+Bcos^2x then prove that sin2x.y_(2)-2cos2x.y_(1)=0 where y_(1)=(dy)/(dx),y_(2)=(d^2y)/(dx^(2)) .

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

If y= cot x, prove that (d^(2)y)/(dx^(2)) + 2y (dy)/(dx)= 0.