Name________________________
Student I.D.___________________
Math 2250-4
Quiz 11 SOLUTIONS
April 12, 2013
1a) Use the methods we've been discussing to find the general solution to the system of differential
equations
x# t
=
y# t
0
1
x
K8 K6
y
.
(8 points)
0Kl
K8
1
K6 K l
2
= l lC6 C8 = l C6 lC8 = lC4
l =K4 the homogeneous system is
4
1 0
K8 K2 0
l =K2 the homogeneous system is
2
1 0
K8 K4 0
/
/
4 1 0
0 0 0
2 1 0
0 0 0
0v=
0v=
lC2
1
K4
1
.
K2
Thus the general solution is
x t
y t
= c1 eK4 t
1
K4
C c2 eK2 t
1
K2
.
1b) For any solution x t , y t T to the first order system of DEs above, what is the second order
differential equation satisfied by x t ?
(2 points)
The first order system reads
x# t = y
y# t =K8 x K 6 y .
Thus x##= y#=K8 x K 6 y =K8 x K 6 x# . We can also write this DE in the usual form for an unforced
undamped mechanical or electrical system
x##C 6 x#C 8 x = 0 .
(And we continued this discussion in class after the quiz, remarking once again on the correspondence
between Chapter 5 and Chapter 7, in the cases where first order systems of DE's arise from converting
nth order linear DE's to systems of n first order linear DE's.)