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Math Help - differential systems

  1. #1
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    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??
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  2. #2
    MHF Contributor

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    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).
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