Math 308
Worksheet 7: Sec 7.5, 7.6, 7.8
25 & 27 July 2012
Key Idea. Take a linear, homogeneous system of first-order differential equations
tx0 = ax + by
ty 0 = cx + dy .
(1a)
Set
dfn
A =
a b
c d
dfn
and X =
x
y
,
and re-write the system (1a) as the single differential (matrix) equation
tX 0 = AX .
(1b)
If V is an eigenvector of A with eigenvalue λ,[i] then
X(t) = tλ V ,
(2)
t > 0,
is one solution of (1).
3
Exercise 1. Suppose that X(t) = t
is one solution of the differential equation
1
tX 0 = AX, with t > 0. Use this information to write outa complete
argument proving
3
that λ = 5 is an eigenvalue of A with eigenvector V =
.
1
5
Back to solving (1): the general solution will be of the form
(3)
X(t) = C1 X1 (t) + C2 X2 (t) ,
t > 0.
The expressions of the solutions {X1 (t) , X2 (t)} depends on the eigenvalues of A.
This is outlined in the summary below.
Solution Summary
I. Distinct real eigenvalues λ1 6= λ2 . Select an eigenvector V1 for λ1 , and an
eigenvector V2 for λ2 . Then X1 (t) = tλ1 V1 and X2 (t) = tλ2 V2 .
Exercises. Use the information given to solve tX 0 = AX.
5
10
1
−3
(2) A
=
and A
=
.
1
2
−2
6
1
5
7
−14
(3) A
=
and A
=
.
3
15
−1
2
II. Complex eigenvalues α ± iβ. In this case, β 6= 0. Select an eigenvector
[i]See
Worksheet 5 for a summary of eigenvalues and (generalized) eigenvectors.
V = R + iS for α + iβ. We obtain the two solutions X1 (t) and X2 (t) in (3) from (2)
by
dfn
X1 (t) + i X2 (t) = tα+iβ V
=
tα cos(ln tβ ) R − sin(ln tβ ) S
+ i tα sin(ln tβ ) R + cos(ln tβ ) S .
Exercises. Use the information given to solve tX 0 = AX.
3
6
6
2
9
4
(4) A
=
−
and A
=
+
.
2
4
3
1
6
2
5
15
2
1
10
3
(5) A
=
−
and A
=
+
.
1
3
4
2
2
6
III. A single eigenvalue λ. Here (as seen in Worksheet 5) there are two possibilities.
C1
[ii]
λ
, then the general solution is X(t) = t
.
Case III.A. If A =
C2
λ 0 [iii]
Case III.B. If A 6=
, then X1 (t) = tλ V . To obtain X2 (t), find a generalized
0 λ
eigenvector W . Then X2 (t) = tλ (W + ln(t)V ) is the second solution.
λ 0
0 λ
Exercises. Use the information given to solve tX 0 = AX.
7
2
6
3
.
=
and A
=
(6) A
4
1
4
2
8
1
15
5
.
=
and A
=
(7) A
7
2
3
1
Homework. As part of your preparation for Quiz 4 (Wednesday, August 01) and
the Final Exam (Tuesday, August 07), I recommend the following:
(A) Section 7.5: Problems 22, 23.
(B) Section 7.6: Problems 13, 15, 17 (a,b); 22.
(C) Section 7.8: Problems 13, 14.
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Rules.
[ii]This
[iii]This
is the case that every vector is an eigenvector.
is the case that the eigenspace E(λ) is a line.
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Colleen Robles
robles@math.tamu.edu