18.034, Honors Differential Equations
Prof. Jason Starr
Lecture 26
4/9/04
1. Spent about ½ lecture working through the linear system of a pair of masses connected
by a spring (of equilibrium displacement L)
m a x a ” = - k(x a - X b - L)
m a x b ” = - k(x a - X b - L)
⎛ xa ⎞
⎜ ⎟
Introduce v a = x a ' , Vb = X b ' , X= ⎜ v a ⎟ ,
⎜ ⎟
⎜ xb ⎟
⎜v ⎟
⎝ b⎠
⎡0
⎢
⎢
⎢− k
A= ⎢ m
= ⎢ a
⎢
⎢0
⎢
⎢ k
⎢
⎣⎢ mb
1
0
0
k
ma
0
0
0 −
k
mb
Then x' = Ax + F . Physics (intuition suggests introducing M =
y1 =
0⎤
⎥
⎥
⎥
0⎥ ,
⎥
⎥
1⎥
⎥
⎥
0⎥
⎦⎥
⎡ 0 ⎤
⎢
⎥
⎢
⎥
kL
⎢
⎥
F= ⎢ ma ⎥ .
= ⎢
⎥
⎢ 0 ⎥
⎢
⎥
⎢ kL ⎥
⎢−
⎥
⎢⎣ mb ⎥⎦
mamb
,
ma + mb
M
M
M
M
xb , y 2 = x a - x b , V1 ' = y 1 ' = Va - Vb . Then
xa +
xa +
xb , V1 = y1' =
ma
mb
mb
ma
⎡0
⎢
⎢
⎢0
y' = By + G where B = ⎢
⎢
⎢0
⎢
⎢
⎢0
⎣
1
0
0
0
0
0
0
−
k
m
0⎤
⎥
⎥
0⎥
⎥ ,
⎥
1⎥
⎥
⎥
0⎥
⎦
⎡0 ⎤
⎢ ⎥
⎢0 ⎥
G = ⎢0 ⎥ .
⎢ ⎥
⎢kL⎥
⎢⎣ M ⎥⎦
Now (y1,V1) is decoupled from (y2,V2).
2. We paused at this point and solved the system by usual 2nd order method. But then we
continued to ‘diagonalize’ (y2,V2).
Implicit C.O.V. :
y 2 = z1 + z 2
V2 = iωz1 + iωz 2
→
⎡iω 0⎤ ⎡ − iωL⎤
⎢
⎥ ⎢ 2M ⎥
⎥ , where ω =
z'= ⎢
⎥z + ⎢
⎢0 − iω⎥ ⎢ + iωL⎥
⎣
⎦ ⎣⎢ 2M ⎦⎥
k
M
This is now completely diagonalized:
⎤
⎡0
⎛ ⎡1 ⎤
⎡y1 ⎤
⎡0⎤
⎡0⎤
⎡1 ⎤
⎡0⎤ ⎞
⎥
⎢
⎟
⎜
⎢ ⎥
0
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎥
⎢
v
0
0
1
0
0
⎟
⎜
1
i
ω
t
−
i
ω
t
⎢ ⎥ = y ⎢ ⎥ +v
⎢ ⎥t + ⎢ ⎥ + z ⎢ ⎥e
+ z2,0 ⎢ ⎥e
+ ⎢L
⎥.
1
,
0
1
,
0
1
,
0
⎟
⎜
⎢ z1 ⎥
⎢1 ⎥
⎢0⎥
⎢0⎥
⎢0⎥
⎢0⎥
⎢ 2m ⎥
⎟
⎜
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎟
⎥
⎢− L
⎜ ⎢0⎥
⎣0⎦
⎣1 ⎦
⎣0⎦
⎣0⎦ ⎠
⎣⎢ z 2 ⎦⎥
⎝⎣ ⎦
⎢⎣ 2m⎥⎦
18.034, Honors Differential Equations
Prof. Jason Starr
Page 1 of 3
Back-substituting:
⎡x a
⎢
⎢v a
⎢x b
⎢
⎢⎣v b
⎤
⎥
⎥ = y
1 ,0
⎥
⎥
⎥⎦
eigenval = 0
eigenvector=
⎛ ⎡1 ⎤
⎡1 ⎤
⎡0 ⎤ ⎞
⎡0 ⎤
⎜⎢ ⎥
⎢ ⎥
⎢ ⎥⎟
⎢ ⎥
⎜ ⎢0 ⎥
⎢0 ⎥ + v
⎢1 ⎥ ⎟ t + ⎢1 ⎥ + z
+
t
1 ,0 ⎜
1 ,0
⎢1 ⎥
⎢ ⎥
⎢0 ⎥ ⎟
⎢0 ⎥
⎜ ⎢1 ⎥
⎢ ⎥
⎢ ⎥ ⎟⎟
⎢ ⎥
⎜
⎣0 ⎦
⎣0 ⎦ ⎠
⎣1 ⎦
⎝ ⎣0 ⎦
eigenval = 0
gen. eigenvector
⎡M
⎢ ma
⎢
M
⎢ iω
ma
⎢
⎢− M
mb
⎢
⎢
M
−
ω
i
⎢
mb
⎣
eigenval = iω
eigenvect
⎤
⎥
⎥
⎥
⎥ e i ω t + z 2 ,0
⎥
⎥
⎥
⎥
⎦
⎡M
⎢ ma
⎢
M
⎢iω
ma
⎢
⎢− M
mb
⎢
⎢
M
ω
i
⎢
mb
⎣
⎤
⎡L
⎥
⎢ ma
⎥
⎢0
⎥
⎥e − ωt + ⎢
⎢− L
⎥
mb
⎢
⎥
⎢0
⎥
⎣
⎥
⎦
eigenval = -iω
eigenvector
Particular sol’n.
Need to choose z1,0 and z2,0 to be complex conjugate to get a real solution.
3. Discussed the general eigenvector decomp. method for solving y' = Ay : If (λ,V) is an
eigenvalue/ eigenvector pair, then y(t) = ve λt is a sol’n.
Defined eigenvalues + eigenvectors.
Defined the char. poly, det(λI - A) .
Saw that the eigenvalues are precisely the roots of det(λI - A) .
We did not yet define/ discuss generalized eigenspaces (although I did mention the term
in the solution of the 2-mass-spring problem).
18.034, Honors Differential Equations
Prof. Jason Starr
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Page 2 of 3
1. One or two problems reducing higher order constant coeff. linear systems to 1st or order
linear systems (students seemed fuzzy on this).
Examples: (a) y" ' - 3y" + 3y'- y = e t :
⎡
⎢0 1
⎢
⎢
y' = ⎢0
0
⎢
⎢0 − 3
⎢
⎣
y1= y
y2= y’
y3= y”
x1 = y1
x2 = y1'
(b)
x3 = y2
y1”=y2
y2”=y3
y3’=y1+y2
x 4 = y2 '
x5 = y3
⎤
0⎥
⎡0 ⎤
⎥
⎢ ⎥
⎥
1⎥ y + ⎢0 ⎥ ,
⎥
⎢e t ⎥
⎣ ⎦
3⎥⎥
⎦
⎡
⎢0 1
⎢
⎢0
0
⎢
⎢
0
x' = ⎢0
⎢
⎢
0
⎢0
⎢
⎢1
0
⎢⎣
0
0
1
0
0
1
0
0
1
0
⎤
0⎥
⎥
0⎥
⎥
⎥
0⎥ x
⎥
⎥
1⎥
⎥
0 ⎥⎥
⎦
2. Maybe one problem going in the other direction:
⎡2 1⎤ ⎡y1 ⎤
'
⎢
⎥⎢ ⎥
⎡y 1 ⎤
y 2 = y 1 '-2y 1
Example: ⎢ ⎥ = ⎢
⎥⎢ ⎥ :
y 1 ” - 2y 1 ' = y 2 ' = y 1 + 2y 1 '-4y 1
⎣y 2 ⎦
⎢1 2⎥ ⎢y ⎥
⎣
⎦⎣ 2 ⎦
y 1 ” - 4y 1 + 3y 1 = 0
y 1 = Ae t + Be 3t
y 2 = -Ae t + Be 3t
3. Several diagonalizing/ eigenvalue- eigenvector problems
⎡− 3 1⎤
⎢
⎥
e.g. (a) ⎢
⎥,
⎢ 1 − 3⎥
⎣
⎦
(b)
⎡1
⎢
⎢
⎢0
⎢
⎢0
⎣
λ2 + 6λ + 8 = (λ + 2)(λ + 4) ; λ = - 2 ,
⎡1⎤
⎢1⎥ ,
⎣⎦
λ = - 4,
⎡ 1⎤
⎢ ⎥
⎢⎣−1⎥⎦
3⎤
⎥
⎥
2 3⎥ .
⎥
0 3⎥
⎦
2
By exercise (18) (This week’s Pset),
⎡1 ⎤
⎢ ⎥
(λ - 1)(λ - 2)(λ - 3) : λ = 1 , 0 ; λ = 2 ,
⎢ ⎥
⎢⎣0⎥⎦
(c)
⎡
⎤
⎢0 − 1⎥
⎢
⎥ , (λ + 1)2 ;
⎢
⎥
⎢1 − 2⎥
⎣
⎦
18.034, Honors Differential Equations
Prof. Jason Starr
⎡2⎤
⎡9 ⎤
⎢1 ⎥ ; λ = 3 , ⎢6 ⎥ .
⎢ ⎥
⎢ ⎥
⎢⎣0 ⎥⎦
⎢⎣2⎥⎦
⎡1⎤
⎣1⎦
λ = -1, ⎢ ⎥ .
Page 3 of 3