MATH 251 Final: MATH 251 PSU s251Final(su11)
Document Summary
7 u8(t) et!8 ! e!6t+48 (a) x(t) = c. "e!9 t (e) proper node (star point), asymptotically stable (a) t (b) f (c) f (d) t (e) t (f) t (a) eigenvalues are = 1, 4, 9, , n2; Eigenfunctions are xn(x) = cos(nx); n = 1, 2, 3, n = 1, 2, 3, (b) yes, 0 is an eigenvalue. X0 = 1, or any nonzero constant function, is an eigenfunction. (b) b n = , n = 1, 2, 3, ; all a"s are 0. (d) constant term: a n = , n = 1, 2, 3, ; all b"s are 0. 1 (e) in part (a), f (4) approaches 0; in part (c), f (4) approaches 1. (a) u(x, t) = n2!2 t. C n e n = 1 sin n!x. 3 (b) u(x, t) = 6e!3!2t sin(!x) +10e!12!2t sin(2!x) !
