If $2x = y^{1/5} + y^{-1/5}$ and $(x^2 - 1) \frac{d^2y}{dx^2} + \lambda x \frac{dy}{dx} + ky = 0$,then $\lambda + k$ is equal to

If $y = \sin(\log_e x)$, then $x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx}$ is equal to

If $x = \log p$ and $y = \frac{1}{p}$,then

The function $y=e^{kx}$ satisfies $(\frac{d^2y}{dx^2}+\frac{dy}{dx})(\frac{dy}{dx}-y)=y\frac{dy}{dx}$. It is valid for

If $p(x)$ is a polynomial of degree $3$ which satisfies $p^{\prime \prime}(1)=0$ and $p^{\prime \prime \prime}(1)=6$,then $p^{\prime \prime}(0)$ is equal to

If ${x^p}{y^q} = {(x + y)^{p + q}}$,then $\frac{{{d^2}y}}{{d{x^2}}} = $