If $y=e^{m \sin ^{-1} x}$, then $(1-x^{2}) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-k y=0$, where $k$ is equal to

If $f(x) = e^{(x+1)^n}; (n \in N)$,then the value of $n$ for which $f''(1) = 67(2^n e^{2^n})$ is

If $y=f(x)$ is a twice differentiable function such that at a point $P$,$\frac{dy}{dx}=4$ and $\frac{d^2y}{dx^2}=-3$,then $\left(\frac{d^2x}{dy^2}\right)_P$ is equal to:

If $\frac{d^n y}{d x^n}=y_n$ and $y=e^{\sqrt{x}}+e^{-\sqrt{x}}$,then $4 x y_2+2 y_1=$

If $f: R \rightarrow R$ is an even function which is twice differentiable on $R$ and $f^{\prime \prime}(\pi)=1$, then $f^{\prime \prime}(-\pi)$ is equal to

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