(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Express the general solutoins of the system of equations in terms of real-valued functions.

x'= [1 0 0; 2 1 -2; 3 2 1]x(I wrote the matrix MATLAB-style)

2. Relevant equations

The coolest equation ever: e^{ib}=cosb +isinb

3. The attempt at a solution

Assumex=Re^{rt}(the underlined r is an eigenvector)

Determinate[1-r 0 0; 2 1-r -2; 3 2 1-r]=0 --> r = 1, 1+2i, 1-2i

r = 1 --> [0 0 0; 2 1 -2; 3 2 0](r_{1}r_{2}r)_{3}^{T}=(0 0 0)^{T}

--->R= (2 -3 2)^{1}^{T}

I do the same with the other eigenvalues, and come up with 3 eigenvectors:R= (2 -3 2)^{1}^{T},R= (0 1 -^{2}i)^{T},R= (0 1^{3}i)^{T}.

By Superpsition, the full solution will be

x(t)=C_{1}e^{t}(2 -3 2)^{T}+ C_{2}e^{t}^{e2it}(0 1 -i)^{T}+ C_{3}e^{t}e^{-2it}(0 1 i)^{T}

= e^{t}[ C_{1}(2 -3 2)^{T}+ C_{2}(cos(2t)+isin(2t))(0 1 -i)^{T}+ C_{3}(cos(-2t)+isin(-2t))(0 1i)^{T}]

................................. This somehow simplifies to the answer in the back of the book, C_{1}e^{t}(2 -3 2)^{T}+ C_{2}e^{t}(0 cos2t sin2t)^{T}+ C_{3}e^{t}(0 sin2t -cos2t)^{T}. I don't understand the simplification process. Yes, I know the imaginary numbers just get absorbed into the constants; but I can't figure out the rest.

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# Solving a system of diffy q's with complex eigenvalues

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