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10Ma2aAnHw5

# 10Ma2aAnHw5 - A are exactly the roots of the polynomial p Î...

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Ma 2a Fall 2010 (analytical track) PROBLEM SET 5 Due on Monday, November 8 I. Undetermined coefficients Problems 16.1(iv), 22.4(v) II. Oscillations Problems 13.4, 13. 8, 15.4 III. Linear systems with constant coefficients Problem 26.1(v) (X1) Rewrite the n -th order linear equation with constant coefficients a 0 x ( n ) + a 1 x ( n - 1) + · · · + a n x = 0 as a linear system ˙ y = Ay in R n . Show that the eigenvalues of the matrix
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Unformatted text preview: A are exactly the roots of the polynomial p ( Î» ) = a Î» n + a 1 Î» n-1 + Â· Â· Â· + a n . (X2) Compute the matrix-valued function e At = âˆž X n-A n n ! t n for A = Â± Î±-Ï‰ Ï‰ Î± Â² and A = Î» 1 Î» 1 Î» . Solve the corresponding IVPs Ë™ x = Ax, x (0) = x . 1...
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