Simpler proof that $y^3[d^2y/dx^2]$ is a constant if $y^2=ax^2+bx+c$? here's my question
If $y^2=ax^2+bx+c$ then prove that  $y^3[d^2y/dx^2]$  is a constant .
I have solved this using the conventional method, taking square root, differentiating w.r.t to x  and using chain and quotient rule
But can't think of some alternative smaller and more efficient method ??
Can you ?
Also can someone suggest me any alternative to quotient rule
I need something more non conventional, time saving and vastly
Applicable method .

Using differentials, it is easy to see that
$$
2y dy = (2ax+ b) dx \tag{1}
$$
from which we can write
$$
u = \frac{dy}{dx} = \frac{2ax+b}{2y}
$$
Using again differentials and product rule,
\begin{eqnarray}
du 
&=& \frac{2a}{2y} dx - \frac{2ax+b}{2y^2} dy 
= \left[ \frac{a}{y} - \frac{(2ax+b)^2}{4y^3} \right] dx 
\end{eqnarray}
where we have used (1).
Finally
\begin{eqnarray}
y^3\frac{du}{dx} 
= y^3\frac{d^2y}{dx^2} 
&=& 
\left[ ay^2 - \frac{(2ax+b)^2}{4} \right]
= 
\left[ a(ax^2+bx+c) - \frac{(2ax+b)^2}{4} \right] 
=
ac-\frac{b^2}{4}
\end{eqnarray}
