MATH 251 Final Exam May 10, 2007 ANSWER KEY 1. C 2. A 3. B 4. D 5. a) Node, asymptotically stable (a “nodal sink”) b) Saddle point, unstable c) Spiral point, unstable (a “spiral source”) d) Center, stable (but not asymptotically stable) e) Node, unstable (a “nodal source”) 6. a) a(x) and e(x) b) b(x) and c(x) 7. −+=22214211)( eettyt 8. a) Eigenvalues are 4)12(22πλ−=n, n = 1, 2, 3, … b) Eigenfunctions are −=xnxXn2)12(cos)(π, n = 1, 2, 3, … 9. a) (1,1), (1,−1), (−1,1), (−1,−1) b) xx−−−=′2222 c) Saddle point, unstable 10. =+′==′=−′+′′00)(,0)0(,075tTTLXXXXXλλ 11. 2510)( −= xxv12. a) 2010)(+=xxv b) )2sin(152sin2010),(22242/3xexextxuttππππ−−+−+= c) 30),1(lim =∞→tut 13. b) π)12(4−−=nbn, n = 1, 2, 3, … c) f (−1) = 1, f (0) = 0, f (3) =
View Full Document
Unlocking...