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Solve the differential equation?

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1 Answer

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:

{e3x, e-3x, e2ix, e-2ix}

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

e2ix = cos 2x + i sin 2x

e-2ix = cos 2x - i sin 2x

(e2ix + e-2ix) / 2 = cos 2x

(e2ix - e-2ix) / 2i = sin 2x

Hence the updated basis is {e3x, e-3x, cos 2x, sin 2x}

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

y = C1 e3x + C2 e-3x + C3 cos 2x + C4 sin 2x

Thus you are correct.