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Unformatted text preview: n Section 2.1 that the eigenspace corresponding
to the other eigenvalue is just the orthogonal complement
W−3 = span −1
1 . Unit length eigenvectors lying in the two eigenspaces are
√
√
1/√2
−1/ 2
√
b1 =
,
b2 =
1/ 2
1/ 2 . The theorem of Section 2.1 guarantees that the matrix
√
√
1/√2 −1/ 2
√
B=
,
1/ 2 1/ 2
whose columns are b1 and b2 , will diagonalize our system of diﬀerential equations. 51 Indeed, if we deﬁne new coordinates (y1 , y2 ) by setting
√
√
x1
1/√2 −1/ 2
y1
√
=
,
x2
y2
1/ 2 1/ 2
our system of diﬀerential equations transforms to
d2 y1 /dt2
d2 y2 /dt2
We set ω1 = 1 and ω2 = √ = −y1 ,
= −3y2 . 3, so that this system assumes the familiar form
2
d2 y1 /dt2 + ω1 y1
2
2
2
d y2 /dt + ω2 y2 =
= 0,
0, a system of two noninteracting harmonic oscillators.
The general solution to the transformed system is
y1 = a1 cos ω1 t + b1 sin ω1 t, y2 = a2 cos ω2 t + b2 sin ω2 t. In the original coordinates, the general solution to (2.15) is
√
√
x1
1/√2 −1/ 2
a1...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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