Let A be an n × n matrix with distinct eigenvalues {λ 1 , ..., λ n } with corresponding linearly independent eigenvectors {v 1 , ..., v n }. Then, the non-homogeneous differential equation x'(t) = Ax(t) + e λ 1 t v 1 :

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Mathematics · Differential Equations

Let A be an n × n matrix with distinct eigenvalues {λ 1 , ..., λ n } with corresponding linearly independent eigenvectors {v 1 , ..., v n }. Then, the non-homogeneous differential equation x'(t) = Ax(t) + e λ 1 t v 1 :

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does not have a solution of the form e λ 1 t a for any vector a &in; &reals; n

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Answer 1

does not have a solution of the form e λ 1 t a for any vector a &in; &reals; n

Answer 2

has a solution of the form e λ 1 t a for some vector a &in; &reals; n

Answer 3

has a solution of the form e λ 1 t a + t e λ 1 t b for some vectors a, b &in; &reals; n

Answer 4

does not have a solution of the form e λ 1 t a + t e λ 1 t b for any vectors a, b &in; &reals; n

Let A be an n × n matrix with distinct eigenvalues {λ 1 , ..., λ n } with corresponding linearly independent eigenvectors {v 1 , ..., v n }. Then, the non-homogeneous differential equation x'(t) = Ax(t) + e λ 1 t v 1 :

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Let A be an n &times; n matrix with distinct eigenvalues {&lambda; 1 , ..., &lambda; n } with corresponding linearly independent eigenvectors {v 1 , ..., v n }. Then, the non-homogeneous differential equation x'(t) = Ax(t) + e &lambda; 1 t v 1 :

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Let A be an n × n matrix with distinct eigenvalues {λ 1 , ..., λ n } with corresponding linearly independent eigenvectors {v 1 , ..., v n }. Then, the non-homogeneous differential equation x'(t) = Ax(t) + e λ 1 t v 1 :