Let's verify it.

You got the characteristic polynomial solved correctly. It indeed factors and has the four roots you listed.

Using the roots we can make a basis.

The initial basis:

{e^{3x}, e^{-3x}, e^{2ix}, e^{-2ix}}

However, the last two solutions involve complex numbers so we must replace them.

e^{2ix} = cos 2x + i sin 2x

e^{-2ix} = cos 2x - i sin 2x

(e^{2ix} + e^{-2ix}) / 2 = cos 2x

(e^{2ix} - e^{-2ix}) / 2i = sin 2x

Hence the updated basis is {e^{3x}, e^{-3x}, cos 2x, sin 2x}

Every solution must be a linear combination of these four so you get the following.

y = C_{1 }e^{3x} + C_{2 }e^{-3x} + C_{3 }
cos 2x + C_{4 }sin 2x

Thus you are correct.

## Comments

Thank you so much.