What Are the Mistakes in Multiplying Eigenvectors for Homogeneous Systems?

[cue928]

In the context of using the eigenvalue method for homogeneous systems, my eigenvector is: V=[1 -i]^T and I have a lambda value of 3-4i. Here's my setup:
(e^3t)[1 -i](cos 4t - i sin 4t)
For the first equation (real):
ME: x1(t) = e^3t(A cos 4t - B sin 4t)
BOOK: x(t) = e^3t(A cos 4t + B sin 4t)

For the second equation (imaginary):
ME: x2(t) = e^3t[cos 4t - i sin 4t - i cos 4t + sin 4t]
BOOK: x2(t) = e^3t[A sin 4t - B cos 4t]

How am I getting the signs wrong on the first equation and what am I not cancelling on the second eq that I should be?

 

[lippyka]

For the first equation, you are getting the signs wrong because you are using a complex eigenvector. The real part of the eigenvector is 1, so the sign should be positive when multiplying it by the cosine term.For the second equation, you are not cancelling out the imaginary terms in the eigenvector with the sin and cos terms. When multiplying an imaginary number times a complex number, you need to ensure that the imaginary terms cancel out. So, for example, multiplying -i * (cos 4t - i sin 4t) should result in i*cos 4t + sin 4t.