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math202assignment06 - (e Show that no values of a and b...

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MATH 202 ASSIGNMENT 6 Please send your solutions via SUCourse as a PDF document attachment. As you all know, we are not accepting any scanned documents. Please try harder to submit your homeworks on time. The deadline is May 29, 2009 11:30 a.m. Show all your work! 1. Consider the second order ODE ( ° ) y 00 + 2 ay 0 + by = 0 ; where a and b are real constants. (a) Show that ( ° ) is equivalent to the system x 0 1 = x 2 ; x 0 2 = ± bx 1 ± 2 ax 2 : (b) Explain why the system has a saddle point if a = 1 and b < 0 and a node if a = 1 and 0 < b < 1 : (c) Show that the system has an improper node if a = b = 1 and a spiral point if a = 1 ; b > 1 . (d) Find all values of a and b for which the system has a center.
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Unformatted text preview: (e) Show that no values of a and b lead to a system that has a proper node. 2. Three solutions of the equation x = Ax are @ e t + e 2 t e 2 t 1 A ; @ e t + e 3 t e 3 t e 3 t 1 A and @ e t ± e 3 t ± e 3 t ± e 3 t 1 A : Find the eigenvalues and eigenvectors of A : 3. Let &; ±; ²; and ³ be constant real numbers and A = @ ³ 1 ³ 1 ³ 1 A : a. Prove that e tA = @ 1 t 1 2 t 2 1 t 1 1 A : b. Use e tA to solve the initial value problem x = Ax ; x (0) = @ & ± ² 1 A : 1...
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