$y=a e^{m x}+b e^{-m x}$ satisfies which of the following differential equations?

$\frac{d^2 y}{d x^2}-m^2 y=0$

We have, $y=a e^{m x}+b e^{-m x}$

$\Rightarrow \frac{d y}{d x}=m a e^{m x}-m b e^{-m x}$

$\Rightarrow \frac{d^2 y}{d x^2}=m^2\left(a e^{m x}+b e^{-m x}\right)$

$\Rightarrow \frac{d^2 y}{d x^2}=m^2 y \Rightarrow \frac{d^2 y}{d x^2}-m^2 y=0$

Hence, option (d) is correct.