Question

If $ y^2 = P (x) $ is a polynomial of degree 3, then $2 \frac{d}{dx} \bigg(y^3 \frac{d^2 \, y}{ dx^2}\bigg) $ equals

• A. P "' (x) + P' (x)
• B. P " (x) - P'" (x)
• C. P (x) P'" (x)
• D. a constant

Answer 1

The Correct Option is C

Since, $ y^2 $ = P (x)
On differentiating both sides, we get
$\hspace16mm$ $ 2 yy_1 $ = P ' (x) ,
Again, differentiating, we get
$ 2yy_2 + 2y_1^2 = $ P " (x)
$\Rightarrow 2y^3 \, y_2 + 2y^2 y_1^2 = y^2 $ P " ( x)
$\Rightarrow 2y^3 y_2 = y^2 \, P "' (x) - 2 (y y_1)^2 $
$\Rightarrow 2y^3 y_2 = P (x) . P " (x) - \frac{ \{ P ' (x) \}^2 }{ 2}$
Again, differentiating, we get
2 $ \frac{d}{dx} (y^3 y_2) = P ' (x) . P " (x) + P (x). P "' (x) - \frac{ 2 P ' (x) . P " (x)}{2}$
$\Rightarrow 2 \frac{d}{dx} (y^3 \, y_2) = P (x). P "' (x)$
$\Rightarrow 2 \frac{d}{dx} \bigg( y^3 . \frac{ d^2 y }{ dx^2} \bigg) = P (x). P "" (x)$