If y2=ax2+bx+c, then y3d2ydx2 is

y2=ax2+bx+cdiff w.r.t. x  ⇒2ydydx=2ax+b⇒dydx=2ax+b2y----(1)diff again w.r.t. x⇒2dydx2+2yd2ydx2=2a⇒yd2ydx2=a-dydx2⇒yd2ydx2=a-2ax+b2y2 from (1)⇒yd2ydx2=4ay2-(2ax+b)24y24y3d2ydx2=4a(y2-ax2-bx)-b24y3d2ydx2=4ac-b2  is a constant  (from given c=y2-ax2-bx)