CLASS NOTES FOR LINEAR MATH

1
CLASS NOTES FOR LINEAR MATH

• Section 5.2
• COMPLEX AND REPEATED ROOTS

2
LEMMA 5.6
L(D)(eat y(t)) eatL(Da)y(t) PROOF
3
LEMMA 5.6
L(D)(eat y(t)) eatL(Da)y(t) PROOF
D(eaty(t)) aeaty(t) eatD(y(t))
eatL(Da)y(t)
4
THEOREM 5.7
If (D - ?)ry 0 then y (c1 c2t ...
crtr-1)e?t PROOF
5
THEOREM 5.7
If (D - ?)ry 0 then y (c1 c2t ...
crtr-1)e?t PROOF Suppose (D - ?)ry 0. Let y
e?tu
6
THEOREM 5.7
If (D - ?)ry 0 then y (c1 c2t ...
crtr-1)e?t PROOF Suppose (D - ?)ry 0. Let y
e?tu then (D - ?)ry (D - ?)re?tu
e?tDru 0
7
THEOREM 5.7
If (D - ?)ry 0 then y (c1 c2t ...
crtr-1)e?t PROOF Suppose (D - ?)ry 0. Let y
e?tu then (D - ?)ry (D - ?)re?tu
e?tDru 0 and since e?t ?
0 then Dru 0
8
THEOREM 5.7
If (D - ?)ry 0 then y (c1 c2t ...
crtr-1)e?t PROOF Suppose (D - ?)ry 0. Let y
e?tu then (D - ?)ry (D - ?)re?tu
e?tDru 0 and since e?t ?
0 then Dru 0 or (integrating r times) u c1
c2t ... crtr-1
9
THEOREM 5.7
If (D - ?)ry 0 then y (c1 c2t ...
crtr-1)e?t PROOF Suppose (D - ?)ry 0. Let y
e?tu then (D - ?)ry (D - ?)re?tu
e?tDru 0 and since e?t ?
0 then Dru 0 or (integrating r times) u c1
c2t ... crtr-1 so y (c1 c2t ...
crtr-1)e?t
10
EXAMPLE 1
Solve
11
EXAMPLE 1
Solve
12
EXAMPLE 1
Solve
13
EXAMPLE 1
Solve
14
THEOREM 5.8
If ((D - a)2 b2)y 0 has eigenvalues ?i
a?bi, then y eat(c1cos bt c2sin bt) PROOF
See pg 252
15
EXAMPLE 2
Solve
16
EXAMPLE 2
Solve
17
EXAMPLE 2
Solve
18
EXAMPLE 2
Solve
19
EXAMPLE 2
Solve
20
EXAMPLE 2
Solve
21
EXAMPLE 2
Solve
or
22
EXAMPLE 2
Solve
or
23
EXAMPLE 2
Solve
or
24
EXAMPLE 2
Solve
or
25
EXAMPLE 2
Solve
or
Notice that, because the solutions are spanning a
subspace of solutions, they are added together.
26
End of Section 5.2