BOUNDED ORBITS OF NONQUASIUNIPOTENT FLOWS ON HOMOGENEOUS ...
MA 303 Homework 1 (Homogeneous Linear Differential …MA 303 Homework 1 (Homogeneous Linear...
Transcript of MA 303 Homework 1 (Homogeneous Linear Differential …MA 303 Homework 1 (Homogeneous Linear...
MA 303Homework 1
(Homogeneous Linear Differential Equations)
Hoon Hong
1st-order, 1 variableProblem:
y'K2 y = 0
y(0) =K3
1. Find the general solution:λK2 = 0
λ = 2
y = C e2 t
2. Find the particular solution:C =K3
y =K3 e2 t
3. Sketch the particular solution:
t
y
t
1 of 48
Problem:y'K2 y = 0
y(0) = 4
1. Find the general solution:λK2 = 0
λ = 2
y = C e2 t
2. Find the particular solution:C = 4
y = 4 e2 t
3. Sketch the particular solution:
t
y
t
2 of 48
Problem:y'C3 y = 0
y(0) = 2
1. Find the general solution:λC3 = 0
λ =K3
y = C eK3 t
2. Find the particular solution:C = 2
y = 2 eK3 t
3. Sketch the particular solution:
t
y
t
3 of 48
Problem:y'C3 y = 0
y(0) =K2
1. Find the general solution:λC3 = 0
λ =K3
y = C eK3 t
2. Find the particular solution:C =K2
y =K2 eK3 t
3. Sketch the particular solution:
t
y
t
4 of 48
2nd-order, 1 variable: Real eigenvaluesProblem:
y''C3 y'C2 y = 0
y(0) =K3
y'(0) = 2
1. Find the general solution:
λ2C3 λC2 = 0
λ =K1, K2
y = C1 eKtCC2 e
K2 t
2. Find the particular solution:
y' =KC1 eKtK2 C2 e
K2 t
C1CC2 =K3
KC1K2 C2 = 2
C1 =K4, C2 = 1
y =K4 eKtCeK2 t
3. Sketch the particular solution:
t
y
t
5 of 48
Problem:y''K3 y'C2 y = 0
y(0) = 3
y'(0) = 1
1. Find the general solution:
λ2K3 λC2 = 0
λ = 2, 1
y = C1 e2 tCC2 e
t
2. Find the particular solution:y' = 2 C1 e
2 tCC2 et
C1CC2 = 3
2 C1CC2 = 1
C1 =K2, C2 = 5
y =K2 e2 tC5 et
3. Sketch the particular solution:
t
y
t
6 of 48
Problem:y''Cy'K6 y = 0
y(0) =K3
y'(0) = 4
1. Find the general solution:
λ2CλK6 = 0
λ = 2, K3
y = C1 e2 tCC2 e
K3 t
2. Find the particular solution:y' = 2 C1 e
2 tK3 C2 eK3 t
C1CC2 =K3
2 C1K3 C2 = 4
C1 =K1, C2 =K2
y =Ke2 tK2 eK3 t
3. Sketch the particular solution:
t
y
t
7 of 48
Problem:y''C5 y'C6 y = 0
y(0) = 3
y'(0) =K8
1. Find the general solution:
λ2C5 λC6 = 0
λ =K2, K3
y = C1 eK2 tCC2 e
K3 t
2. Find the particular solution:y' =K2 C1 e
K2 tK3 C2 eK3 t
C1CC2 = 3
K2 C1K3 C2 =K8
C1 = 1, C2 = 2
y = eK2 tC2 eK3 t
3. Sketch the particular solution:
t
y
t
8 of 48
Problem:y''C7 y'C12 y = 0
y(0) = 6
y'(0) =K22
1. Find the general solution:
λ2C7 λC12 = 0
λ =K3, K4
y = C1 eK3 tCC2 e
K4 t
2. Find the particular solution:y' =K3 C1 e
K3 tK4 C2 eK4 t
C1CC2 = 6
K3 C1K4 C2 =K22
C1 = 2, C2 = 4
y = 2 eK3 tC4 eK4 t
3. Sketch the particular solution:
t
y
t
9 of 48
Problem:y''K5 y'C6 y = 0
y(0) = 2
y'(0) = 1
1. Find the general solution:
λ2K5 λC6 = 0
λ = 3, 2
y = C1 e3 tCC2 e
2 t
2. Find the particular solution:y' = 3 C1 e
3 tC2 C2 e2 t
C1CC2 = 2
3 C1C2 C2 = 1
C1 =K3, C2 = 5
y =K3 e3 tC5 e2 t
3. Sketch the particular solution:
t
y
t
10 of 48
Problem:y''Cy'K2 y = 0
y(0) =K4
y'(0) = 5
1. Find the general solution:
λ2CλK2 = 0
λ = 1, K2
y = C1 etCC2 e
K2 t
2. Find the particular solution:y' = C1 e
tK2 C2 eK2 t
C1CC2 =K4
C1K2 C2 = 5
C1 =K1, C2 =K3
y =KetK3 eK2 t
3. Sketch the particular solution:
t
y
t
11 of 48
Problem:y''C5 y'C6 y = 0
y(0) =K1
y'(0) = 1
1. Find the general solution:
λ2C5 λC6 = 0
λ =K2, K3
y = C1 eK2 tCC2 e
K3 t
2. Find the particular solution:y' =K2 C1 e
K2 tK3 C2 eK3 t
C1CC2 =K1
K2 C1K3 C2 = 1
C1 =K2, C2 = 1
y =K2 eK2 tCeK3 t
3. Sketch the particular solution:
t
y
t
12 of 48
2nd-order, 1 variable: Non-real eigenvaluesProblem:
y''K4 y'C68 y = 0
y(0) = 1
y'(0) = 10
1. Find the general solution:
λ2K4 λC68 = 0
λ = 2C8 I, 2K8 I
y = e2 t C1 cos 8 t CC2 sin 8 t
2. Find the particular solution:
y' = 2 e2 t C1 cos 8 t CC2 sin 8 t Ce2 t K8 C1 sin 8 t C8 C2 cos 8 t
C1 = 1
2 C1C8 C2 = 10
C2 = 1
y = e2 t cos 8 t Csin 8 t
3. Sketch the particular solution:
t
y
t
13 of 48
Problem:y''C2 y'C50 y = 0
y(0) = 2
y'(0) = 19
1. Find the general solution:
λ2C2 λC50 = 0
λ =K1C7 I, K1K7 I
y = eKt C1 cos 7 t CC2 sin 7 t
2. Find the particular solution:y' =KeKt C1 cos 7 t CC2 sin 7 t CeKt K7 C1 sin 7 t C7 C2 cos 7 t
C1 = 2
KC1C7 C2 = 19
C2 = 3
y = eKt 2 cos 7 t C3 sin 7 t
3. Sketch the particular solution:
t
y
t
14 of 48
Problem:y''C64 y = 0
y(0) = 2
y'(0) = 8
1. Find the general solution:
λ2C64 = 0
λ = 8 I, K8 I
y = C1 cos 8 t CC2 sin 8 t
2. Find the particular solution:y' =K8 C1 sin 8 t C8 C2 cos 8 t
C1 = 2
8 C2 = 8
C2 = 1
y = 2 cos 8 t Csin 8 t
3. Sketch the particular solution:
t
y
t
15 of 48
Problem:y''C2 y'C82 y = 0
y(0) = 0
y'(0) = 9
1. Find the general solution:
λ2C2 λC82 = 0
λ =K1C9 I, K1K9 I
y = eKt C1 cos 9 t CC2 sin 9 t
2. Find the particular solution:y' =KeKt C1 cos 9 t CC2 sin 9 t CeKt K9 C1 sin 9 t C9 C2 cos 9 t
C1 = 0
KC1C9 C2 = 9
C2 = 1
y = eKt sin 9 t
3. Sketch the particular solution:
t
y
t
16 of 48
Problem:y''K4 y'C125 y = 0
y(0) = 1
y'(0) = 35
1. Find the general solution:
λ2K4 λC125 = 0
λ = 2C11 I, 2K11 I
y = e2 t C1 cos 11 t CC2 sin 11 t
2. Find the particular solution:y' = 2 e2 t C1 cos 11 t CC2 sin 11 t Ce2 t K11 C1 sin 11 t C11 C2 cos 11 t
C1 = 1
2 C1C11 C2 = 35
C2 = 3
y = e2 t cos 11 t C3 sin 11 t
3. Sketch the particular solution:
t
y
t
17 of 48
Problem:y''C9 y = 0
y(0) = 1
y'(0) =K3
1. Find the general solution:
λ2C9 = 0
λ = 3 I, K3 I
y = C1 cos 3 t CC2 sin 3 t
2. Find the particular solution:y' =K3 C1 sin 3 t C3 C2 cos 3 t
C1 = 1
3 C2 =K3
C2 =K1
y = cos 3 t Ksin 3 t
3. Sketch the particular solution:
t
y
t
18 of 48
1st-order, 2 variables: IntroductionProblem:
y1' =Ky1C2 y2
y2' = 2 y1C2 y2
y1(0) = 3
y2(0) = 1
1. Find the general solution:K1Kλ v1C2 v2 = 0
2 v1C 2Kλ v2 = 0
λ2KλK6 = 0
λ = 3
v1 = 2 C1
v2 = 4 C1
λ =K2
v1 = 2 C2
v2 =KC2
y1 = 2 C1 e3 tC2 C2 e
K2 t
y2 = 4 C1 e3 tKC2 e
K2 t
2. Find the particular solution:2 C1C2 C2 = 3
4 C1KC2 = 1
C1 =12
, C2 = 1
y1 = e3 tC2 eK2 t
y2 = 2 e3 tKeK2 t
19 of 48
Problem:y1' = y1K2 y2
y2' =K2 y1K2 y2
y1(0) = 3
y2(0) = 1
1. Find the general solution:1Kλ v1K2 v2 = 0
K2 v1C K2Kλ v2 = 0
λ2CλK6 = 0
λ =K3
v1 =K2 C1
v2 =K4 C1
λ = 2
v1 =K2 C2
v2 = C2
y1 =K2 C1 eK3 tK2 C2 e
2 t
y2 =K4 C1 eK3 tCC2 e
2 t
2. Find the particular solution:K2 C1K2 C2 = 3
K4 C1CC2 = 1
C1 =K12
, C2 =K1
y1 = eK3 tC2 e2 t
y2 = 2 eK3 tKe2 t
20 of 48
Problem:y1' =K5 y1K2 y2
y2' = 2 y1
y1(0) = 1
y2(0) = 1
1. Find the general solution:K5Kλ v1K2 v2 = 0
Kλ v2C2 v1 = 0
λ2C5 λC4 = 0
λ =K1
v1 =K2 C1
v2 = 4 C1
λ =K4
v1 =K2 C2
v2 = C2
y1 =K2 C1 eKtK2 C2 e
K4 t
y2 = 4 C1 eKtCC2 e
K4 t
2. Find the particular solution:K2 C1K2 C2 = 1
4 C1CC2 = 1
C1 =12
, C2 =K1
y1 =KeKtC2 eK4 t
y2 = 2 eKtKeK4 t
21 of 48
Problem:y1' = 3 y1Ky2
y2' =K2 y1C4 y2
y1(0) = 1
y2(0) = 4
1. Find the general solution:3Kλ v1Kv2 = 0
K2 v1C 4Kλ v2 = 0
λ2K7 λC10 = 0
λ = 2
v1 =KC1
v2 =KC1
λ = 5
v1 =KC2
v2 = 2 C2
y1 =KC1 e2 tKC2 e
5 t
y2 =KC1 e2 tC2 C2 e
5 t
2. Find the particular solution:KC1KC2 = 1
KC1C2 C2 = 4
C1 =K2, C2 = 1
y1 = 2 e2 tKe5 t
y2 = 2 e2 tC2 e5 t
22 of 48
1st-order, 2 variables: Real eigenvalues. Opposite SignsProblem:
y' =K1 2
2 2 y
y(0) =3
1
1. Find the general solution:
Det K1Kλ 2
2 2Kλ= 0
λ2KλK6 = 0
λ = 3, v =2
4
λ =K2, v =2
K1
y = C1 2
4 e3 tCC2
2
K1 eK2 t
2. Sketch the general solution:
y1
y2
Saddle
3. Find the particular solution:
23 of 48
C1 2
4CC2
2
K1=
3
1
C1 =12
, C2 = 1
y =1
2 e3 tC
2
K1 eK2 t
4. Sketch the particular solution:
t
y
t
24 of 48
Problem:
y' =1 K2
K2 K2 y
y(0) =3
1
1. Find the general solution:
Det 1Kλ K2
K2 K2Kλ= 0
λ2CλK6 = 0
λ =K3, v =K2
K4
λ = 2, v =K2
1
y = C1 K2
K4 eK3 tCC2
K2
1 e2 t
2. Sketch the general solution:
y1
y2
Saddle
3. Find the particular solution:
C1 K2
K4CC2
K2
1=
3
1
C1 =K12
, C2 =K1
25 of 48
Problem:
y' =K2 2
2 1 y
y(0) =2
K6
1. Find the general solution:
Det K2Kλ 2
2 1Kλ= 0
λ2CλK6 = 0
λ = 2, v =2
4
λ =K3, v =2
K1
y = C1 2
4 e2 tCC2
2
K1 eK3 t
2. Sketch the general solution:
y1
y2
Saddle
3. Find the particular solution:
C1 2
4CC2
2
K1=
2
K6
C1 =K1, C2 = 2
27 of 48
1st-order, 2 variables: Real eigenvalues. Same SignsProblem:
y' =3 K1
K2 4 y
y(0) =1
4
1. Find the general solution:
Det 3Kλ K1
K2 4Kλ= 0
λ2K7 λC10 = 0
λ = 2, v =K1
K1
λ = 5, v =K1
2
y = C1 K1
K1 e2 tCC2
K1
2 e5 t
2. Sketch the general solution:
y1
y2
Unstable
3. Find the particular solution:
29 of 48
C1 K1
K1CC2
K1
2=
1
4
C1 =K2, C2 = 1
y =2
2 e2 tC
K1
2 e5 t
4. Sketch the particular solution:
t
y
t
30 of 48
Problem:
y' =K3 2
1 K2 y
y(0) =8
2
1. Find the general solution:
Det K3Kλ 2
1 K2Kλ= 0
λ2C5 λC4 = 0
λ =K1, v =2
2
λ =K4, v =2
K1
y = C1 2
2 eKtCC2
2
K1 eK4 t
2. Sketch the general solution:
y1
y2
Stable
3. Find the particular solution:
C1 2
2CC2
2
K1=
8
2
C1 = 2, C2 = 2
31 of 48
Problem:
y' =2 1
2 3 y
y(0) =0
3
1. Find the general solution:
Det 2Kλ 1
2 3Kλ= 0
λ2K5 λC4 = 0
λ = 1, v =1
K1
λ = 4, v =1
2
y = C1 1
K1 etCC2
1
2 e4 t
2. Sketch the general solution:
y1
y2
Unstable
3. Find the particular solution:
C1 1
K1CC2
1
2=
0
3
C1 =K1, C2 = 1
33 of 48
Problem:
y' =2 K1
K1 2 y
y(0) =K3
1
1. Find the general solution:
Det 2Kλ K1
K1 2Kλ= 0
λ2K4 λC3 = 0
λ = 1, v =K1
K1
λ = 3, v =K1
1
y = C1 K1
K1 etCC2
K1
1 e3 t
2. Sketch the general solution:
y1
y2
Unstable
3. Find the particular solution:
C1 K1
K1CC2
K1
1=
K3
1
C1 = 1, C2 = 2
35 of 48
1st-order 2 variables: Non-real eigenvaluesProblem:
y' =K2 K10
10 K2 y
y(0) =1
K1
1. Find the general solution:
Det K2Kλ K10
10 K2Kλ= 0
λ2C4 λC104 = 0
λ =K2C10 I, v =K10
10 I
λ =K2K10 I, v =K10
K10 I
α =K2, β = 10, p =K10
0, q =
0
10
y = eK2 t C1 K10
0CC2
0
10 cos 10 t C C2
K10
0KC1
0
10 sin 10 t
2. Sketch the general solution:
y1
y2
Stable
37 of 48
3. Find the particular solution:
C1 K10
0CC2
0
10=
1
K1
C1 =K110
, C2 =K110
y = eK2 t 1
K1 cos 10 t C
1
1 sin 10 t
4. Sketch the particular solution:
t
y
t
38 of 48
Problem:
y' =K7 16
K8 9 y
y(0) =4
K4
1. Find the general solution:
Det K7Kλ 16
K8 9Kλ= 0
λ2K2 λC65 = 0
λ = 1C8 I, v =16
8C8 I
λ = 1K8 I, v =16
8K8 I
α = 1, β = 8, p =16
8, q =
0
8
y = et C1 16
8CC2
0
8 cos 8 t C C2
16
8KC1
0
8 sin 8 t
2. Sketch the general solution:
y1
y2
Unstable
3. Find the particular solution:
39 of 48
C1 16
8CC2
0
8=
4
K4
C1 =14
, C2 =K34
y = et 4
K4 cos 8 t C
K12
K8 sin 8 t
4. Sketch the particular solution:
t
y
t
40 of 48
Problem:
y' =2 10
K4 K2 y
y(0) =1
1
1. Find the general solution:
Det 2Kλ 10
K4 K2Kλ= 0
λ2C36 = 0
λ = 6 I, v =10
K2C6 I
λ =K6 I, v =10
K2K6 I
α = 0, β = 6, p =10
K2, q =
0
6
y = C1 10
K2CC2
0
6 cos 6 t C C2
10
K2KC1
0
6 sin 6 t
2. Sketch the general solution:
y1
y2
center
3. Find the particular solution:
41 of 48
C1 10
K2CC2
0
6=
1
1
C1 =110
, C2 =15
y =1
1 cos 6 t C
2
K1 sin 6 t
4. Sketch the particular solution:
t
y
t
42 of 48
Problem:
y' =1 K4
4 1 y
y(0) =1
K1
1. Find the general solution:
Det 1Kλ K4
4 1Kλ= 0
λ2K2 λC17 = 0
λ = 1C4 I, v =K4
4 I
λ = 1K4 I, v =K4
K4 I
α = 1, β = 4, p =K4
0, q =
0
4
y = et C1 K4
0CC2
0
4 cos 4 t C C2
K4
0KC1
0
4 sin 4 t
2. Sketch the general solution:
y1
y2
Unstable
3. Find the particular solution:
43 of 48
C1 K4
0CC2
0
4=
1
K1
C1 =K14
, C2 =K14
y = et 1
K1 cos 4 t C
1
1 sin 4 t
4. Sketch the particular solution:
t
y
t
44 of 48
Problem:
y' =K14 24
K12 10 y
y(0) =4
K4
1. Find the general solution:
Det K14Kλ 24
K12 10Kλ= 0
λ2C4 λC148 = 0
λ =K2C12 I, v =24
12C12 I
λ =K2K12 I, v =24
12K12 I
α =K2, β = 12, p =24
12, q =
0
12
y = eK2 t C1 24
12CC2
0
12 cos 12 t C C2
24
12KC1
0
12 sin 12 t
2. Sketch the general solution:
y1
y2
Stable
3. Find the particular solution:
45 of 48
C1 24
12CC2
0
12=
4
K4
C1 =16
, C2 =K12
y = eK2 t 4
K4 cos 12 t C
K12
K8 sin 12 t
4. Sketch the particular solution:
t
y
t
46 of 48
Problem:
y' =6 K10
4 K6 y
y(0) =2
1
1. Find the general solution:
Det 6Kλ K10
4 K6Kλ= 0
λ2C4 = 0
λ = 2 I, v =K10
K6C2 I
λ =K2 I, v =K10
K6K2 I
α = 0, β = 2, p =K10
K6, q =
0
2
y = C1 K10
K6CC2
0
2 cos 2 t C C2
K10
K6KC1
0
2 sin 2 t
2. Sketch the general solution:
y1
y2
center
3. Find the particular solution:
47 of 48