Name________________________
Student I.D.___________________
Math 2250-4
Quiz 12 Solutions
April 20, 2012
1) Find the general solution x1 t , x2 t T to the homogeneous system of second order differential
equations, which could result from two masses coupled together with springs:
x1 ## t =K3 x1 C x2
x2 ## t = 2 x1 K 2 x2 .
x1 ## t
x2 ## t
K3 K l
2
1
K2 K l
=
K3 1
2 K2
(8 points)
x1 t
x2 t
2
= lC 3 lC 2 K 2 = l C 5 lC 4
= lC 4 lC 1 .
l =K1 w =
Kl = 1
the homogeneous system is
K2 1 0
K2 1 0
1
/
0v=
2 K1 0
0 0 0
2
.
l =K4 w = 2 the homogeneous system is
1 1 0
1 1 0
1
/
0v=
2 2 0
0 0 0
K1
.
So,
x1 t
x2 t
= c1cos t C c2sin t
1
C c3cos 2 t C c4sin 2 t
2
1
K1
.
2) Use Newton's second law to derive the system of differential equations above, from the configuration
below. x1 t , x2 t are the displacements from equilibrium of two masses. The mass values and spring
constants are as shown.
(2 points)
2 x1 ## t =K4 x1 C 2 x2 K x1 =K6 x1 C 2 x2
1 x1 ## t =K2 x2 K x1 = 2 x1 K 2 x2 .
This is equivalent to the system in problem 1 (divide the first DE by 2).