An ellipse whose center is at origin and the axes are the coordinate axis is represented by the equation.
x2a2+y2b2=1 where a>b -(i)
Differentiating (i) with respect to x once we obtain
2xa2+2yy′b2=0 where y′=dydx -(ii)
Differentiating (ii) with respect to x we obtain
2a2+2[(y′)2+y′′b2]=0 where y′′=d2ydx2
⇒b2a2=−[(y′)2+y′′]1 -(iv)
Consider equation (ii)
2xa2+2yy′b2=0⇒xa2+yy′b2=0⇒b2a2=−yy′x -(v)
Equating (iv) & (v) we have
yy′x=y′2+y′′⇒yy′=x[(y′)2+y′′]⇒ydydx=x[(dydx)2+yd2ydx2]