Higher-order differential equations, the Laplace transform and exponential order

Dear Friends,

I see so many books that show the Laplace transform and its application to linear differential equations for second-order and/or higher-order differential equations.

But as we know the Laplace transform can be applied when the solutions are known to be of exponential order, and for first-order equations we indeed know it by the Gronwall's inequality, but what about higher-order equations?

For instance,

$\displaystyle y^{(n+1)}(t)+py(t)=0\quad\text{for}\ t\geq0,$

where $\displaystyle n\in\mathbb{N}$ and $\displaystyle p\in\mathbb{R}$, do we know that the solutions are of exponential order?

Thanks!