
differential systems
when solving a system of differential equations of the form
dy/dt = Ay
The constant vector lambda is found with
A  lambda * I = 0
If lambda is a complex number I have the solution to
A = [4, 2; 5, 2]
with lambda = 3 + 3i
y = A Re {[2; 13i]exp(3+3i)t} + B Im {[2; 13i]exp(3+3i)t} (1)
= A exp(3t)[2cos3t; cos3t + 3sin3t] + B exp(3t)[2sin3t; sin3t  3cos3t] (2)
The question, how do i get from equation (1) to (2)???
does the imaginary part equal sin and the real part equal cos??

$\displaystyle e^{ibt}= cos(bt)+ i sin(bt)$ and $\displaystyle e^{ibt}= cos(bt) i sin(bt)$ so that
$\displaystyle C_1e^{a+ ibt}+ C_2e^{aibt}= C_1e^{at}(cos(bt)+ i sin(bt))+ C_2e^{at}(cos(bt) i sin(bt))$
$\displaystyle = e^{at}\left((C_1+ C_2)cos(bt)+ i(C_1 C_2) sin(bt)\right)$
$\displaystyle = e^{at}\left(D_1 cos(bt)+ D_2 sin(bt)\right)$
where $\displaystyle D_1= C_1+ C_2$ and $\displaystyle D_2= i(C_1 C_2)$.