Name________________________
Student I.D.___________________
Math 2250-1
Quiz 12 Solutions
November 30, 2012
Here is a matrix, and its eigenvalue-eigenvector data:
K3
A :=
For l1 =K1, v =
2
2 K3
1
1
. For l2 =K5, v =
1
K1
.
1) Consider the following first order system of first order differential equations, which could arise from
an input-output model:
x1 # t =K3 x1 C 2 x2
x2 # t = 2 x1 K 3 x2 .
Use the eigendata you have been given to write down the general solution to this system of DE's.
(4 points)
x1 t
x2 t
= c1 eKt
1
1
C c2 eK5 t
1
K1
.
2) Consider the following system of second order differential equations, which could arise from an
undamped configuration of two masses and three springs connected in parallel, as we've considered this
week.
x1 ## t =K3 x1 C 2 x2
x2 ## t = 2 x1 K 3 x2 .
a) Use the eigendata you have been given to write down the general solution to this system of DE's.
(4 points)
l =K1 0 w = 1; l =K5 0 w = 5 so,
x1 t
x2 t
= c1 cos t C c2 sin t
1
1
C c3 cos
5 t C c4 sin
5t
1
K1
.
b) Solve the initial value problem for this system of DE's, with initial data
x1 0 = 2, x1 # 0 = 0, x2 0 = 2, x2 # 0 = 0 .
(2 points)
x1 0 = 2 = c1 C c3
x2 0 = 2 = c1 K c3
x1 # 0 = 0 = c2 C
5 c4
x2 # 0 = 0 = c2 K 5 c4 .
So c1 = 2, c3 = 0, c2 = 0, c4 = 0 .