# differential systems

• Apr 23rd 2009, 11:39 AM
revolution2000
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; 1-3i]exp(3+3i)t} + B Im {[2; 1-3i]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??
• Apr 25th 2009, 01:23 AM
HallsofIvy
$e^{ibt}= cos(bt)+ i sin(bt)$ and $e^{-ibt}= cos(bt)- i sin(bt)$ so that
$C_1e^{a+ ibt}+ C_2e^{a-ibt}= C_1e^{at}(cos(bt)+ i sin(bt))+ C_2e^{at}(cos(bt)- i sin(bt))$

$= e^{at}\left((C_1+ C_2)cos(bt)+ i(C_1- C_2) sin(bt)\right)$

$= e^{at}\left(D_1 cos(bt)+ D_2 sin(bt)\right)$

where $D_1= C_1+ C_2$ and $D_2= i(C_1- C_2)$.