Answer to Question #337018 in Differential Equations for Rayd

Question

Suppose 𝐴 is real 3 × 3 matrix that has the following eigenvalues and eigenvectors: −2, ( 1 1 1 ) , 1 + 𝑖, ( 1 − 𝑖 2 1 ) , 1 − 𝑖, ( 1 + 𝑖 2 1 ). Find a fundamental set of real valued solutions to 𝐱 ′ = 𝐴𝐱.

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Answer: The first eigenvalue/eigenvector pair gives the solution 𝐱𝟏 (𝑡) = ( 1 1 1 ) 𝑒^ −2𝑡 . The second eigenvalue/eigenvector pair gives the two solutions: 𝐱𝟐 (𝑡) = ( cos(𝑡) + sin(𝑡) 2 cos(𝑡) cos(𝑡) )𝑒^ 𝑡 , 𝐱𝟑 (𝑡) = ( − cos(𝑡) + sin(𝑡) 2 sin(𝑡) sin(𝑡) )𝑒^ t

Answer 1

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