If $y = a{e^{mx}} + b{e^{ - mx}}$,then $\frac{d^2y}{dx^2} - m^2y = $

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

For $y=\sin ^{-1}\left\{\frac{5 x+12 \sqrt{1-x^{2}}}{13}\right\} ;|x| \leq 1$,if $a\left(1-x^{2}\right) y_{2}+b x y_{1}=0$ then $(a, b)=$

If $y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \dots \infty}}}$,then the value of $\frac{d^2 y}{d x^2}$ at the point $(\pi, 1)$ is

If $y = (ax + b) \cos x$,then $y_2 + y_1 \sin 2x + y(1 + \sin^2 x) = $

If $y=a \sin x+(5+2 x) \cos x$,then $y^{\prime \prime}+y=$