In order to prove that solutions of linear equations with constant coefficients grow at most exponentially, you can use a multidimensional version of Gronwall's lemma.

Remember indeed that the solution

of a

-th order linear differential equation is a component of the solution

of a

-dimensional linear equation. Then an exponential bound on the euclidean norm of

gives an exponential bound on

. Here's how to obtain such a bound.

If

where

are

-matrices, we have

for every

, so that

, hence

. And now you can apply the one-dimensional Gronwall lemma "in integral form" (quoting the

wikipedia) to the function

. The only online reference I could find for the final result is this french

exercise sheet (Exercice 1.4), but you don't need it to find the proof, just apply Gronwall's lemma like I wrote and do an integration by parts to compute the integral. You should get:

.

This is what you need.