Unformatted text preview: nodal source
+
1 + 2 3 < a < 3—unstable node = nodal source
a = 3—unstable degenerate comb
3 < a—saddle (b)(c) [18] Here are�pictures for a = 0, 1, 2, −1 +
�
2 3, 2.75, 3, 4.�(a = −2 3 omitted.) The picture for some
a < −1 − 2 3 would show a nodal sink, and that for
�
a = −1 − 2 3 would show a defective nodal sink. � �
a −b
36. (a) [9] With A =
, pA (�) = �2 − 2a� +(a2 + b2 ) = (� − a)2 + b2 , so the eigenvalues
ba
�
�
−bi −b
are a ± bi. An eigenvector for �1 = a + bi is given by v1 such that
v1 = 0, and
b −bi
�
�
�
�
1
1
we can take v1 =
. The corresponding normal mode is e(a+bi)t
. Its real and
−i
−i
�
�
�
�
cos(bt)
sin(bt)
at
at
imaginary parts give linearly independent real solutions, e
and e
.
sin(bt)
cos(bt) �
�
�
cos(bt) sin(bt)
1...
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This note was uploaded on 01/22/2014 for the course MATH 18.03 taught by Professor Vogan during the Spring '09 term at MIT.
 Spring '09
 vogan
 Differential Equations, Equations

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