If $y = \frac{A}{x} + B x^2$, then $x^2 \frac{d^2 y}{d x^2} =$

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

If $f(x)$ and $g(x)$ are twice differentiable functions on $(0,3)$ satisfying $f^{\prime \prime}(x)=g^{\prime \prime}(x)$, $f^{\prime}(1)=4$, $g^{\prime}(1)=6$, $f(2)=3$, and $g(2)=9$, then $f(1)-g(1)$ is

If $y = {\left[ {x + \sqrt {{x^2} - 1} } \right]^{15}} + {\left[ {x - \sqrt {{x^2} - 1} } \right]^{15}}$,then $\left( {{x^2} - 1} \right)\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}}$ is equal to

If $y = (\sin^{-1} 2x)^2 + (\cos^{-1} 2x)^2$, then $(1 - 4x^2) y_2 - 4x y_1 = $

If $x = \sqrt{e^{\sin^{-1} t}}$ and $y = \sqrt{e^{\cos^{-1} t}}$,then $\frac{d^2 y}{dx^2}$ is