18.034, Honors Differential Equations
Prof. Jason Starr
Recitation Suggestions
4/14/04
Next time I will prove the generalized eigenspaces give a direct sum decomposition. Then I will
explain Jordan normal form and how this gives the complete solution of a linear system of 1st
order ODE’s over Cl, (and then explain what needs to done /IR ).
Until then, please keep doing examples of finding generalized eigenvectors and using the usual
eigenvectors in simple cases (i.e. when generalized eigenspaces = eigenspaces) to solve
systems of 1st order ODE’s.
1
Examples: 1
λ = a + bi :
⎛a
b ⎞⎟
⎜
⎟y .
y' = ⎜
⎟
⎜
⎜ − b a⎟
⎠
⎝
⎛1 ⎞
v = ⎜⎜ ⎟⎟ ;
⎝i ⎠
λ = a − bi :
p A (λ ) = (λ - a)2 + b 2 ,
λ = a ± bi
⎛ ⎞
⎜ 1⎟
v = ⎜ ⎟.
⎜ ⎟
⎝− i⎠
⎛ ⎞
⎜ 1⎟
⎛1⎞ at ibt
⎟
⎜
y = A⎜ ⎟e e + B⎜ ⎟e at e − ibt
⎝i ⎠
⎜ ⎟
⎝− i⎠
⎛ sin(bt ) ⎞ at
⎛ cos(bt ) ⎞ at
⎟⎟e
⎟⎟e + C 2 ⎜⎜
y = C1 ⎜⎜
⎝ cos(bt )⎠
⎝ − sin(bt )⎠
Xa and Xb are “reduced” coords.
incorporating natural equilibrium
displacement of the springs.
⎡0
1 0 0⎤
⎥
⎢
⎥ ⎡x ⎤
⎛ xa ⎞ ⎢ − 2k
a
k
0
0 ⎥ ⎢v ⎥
mxa " = −kx a + k( xb − x a ) v a = x a ' ⎜⎜ v ⎟⎟ ⎢⎢
⎥ ⎢ a⎥
, a
m
m
⎥ ⎢x ⎥
mxb " = −k(xb − x a )
v b = xb ' ⎜ x ⎟ = ⎢
⎜ b ⎟ ⎢0
0 0 1⎥ ⎢ b ⎥
⎜v ⎟ ⎢
⎥ ⎢⎣vb ⎥⎦
⎝ b⎠
⎥
⎢
k
k
⎢
0
0⎥
⎥⎦
⎢⎣ m
m
2
2
2
⎛k ⎞
⎛k ⎞
⎛k ⎞
Tr = 0 , det = ⎜ ⎟ , PA (λ ) = λ 4 + 3⎜ ⎟ λ2 + ⎜ ⎟ ,
⎝m⎠
⎝ m⎠
⎝ m⎠
⎛k⎞
⎟ , then
⎝m⎠
ω2 = ⎜
⎡2
⎢
⎛3 ± s ⎞ k
3+ s
3− s
⎢2λ
⎟
λ = ±i
ω, ±i
ω . If λ2 = −⎜
,
then
eigenvector
is
⎢1 ± s
⎜ 2 ⎟m
2
2
⎝
⎠
⎢
⎢⎣ 1 ± s λ
(
⎤
⎥
⎥
⎥.
⎥
⎥⎦
)
Interesting result: For the “normal modes” x b and x a are either in phase or 180° out of phase.
18.034, Honors Differential Equations
Prof. Jason Starr
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