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Consider a differential equation * = Ax + b(t) (*) with x ER" and b: R → Rn continuous and of period 7 > 0, that is, b(t + 7) = b(t) for some 7 > 0. (a) Assume the following condition on the matrix A holds: XT If X is an eigenvalue of A then is not an integer. Σπί Prove that equation (*) has a unique solution of period 7. (**) (b) Prove that if (**) is false then there exists some b(t) as above such that equation (*) does not have a T-periodic solution.

Consider a differential equation * = Ax + b(t) (*) with x ER" and b: R → Rn continuous and of period 7 > 0, that is, b(t + 7) = b(t) for some 7 > 0. (a) Assume the following condition on the matrix A holds: XT If X is an eigenvalue of A then is not an integer. Σπί Prove that equation (*) has a unique solution of period 7. (**) (b) Prove that if (**) is false then there exists some b(t) as above such that equation (*) does not have a T-periodic solution.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider a differential equation * = Ax + b(t) (*) with x ER" and b: R → Rn continuous and of period 7 > 0, that is, b(t + 7) = b(t) for some 7 > 0. (a) Assume the following condition on the matrix A holds: XT If X is an eigenvalue of A then is not an integer. Σπί Prove that equation (*) has a unique solution of period 7. (**) (b) Prove that if (**) is false then there exists some b(t) as above such that equation (*) does not have a T-periodic solution.
Transcribed Image Text:Consider a differential equation * = Ax + b(t) (*) with x ER" and b: R → Rn continuous and of period 7 > 0, that is, b(t + 7) = b(t) for some 7 > 0. (a) Assume the following condition on the matrix A holds: XT If X is an eigenvalue of A then is not an integer. Σπί Prove that equation (*) has a unique solution of period 7. (**) (b) Prove that if (**) is false then there exists some b(t) as above such that equation (*) does not have a T-periodic solution.
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