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Transcript of MA 303 Homework 1 (Homogeneous Linear Differential MA 303 Homework 1 (Homogeneous Linear...

• MA 303 Homework 1

(Homogeneous Linear Differential Equations)

Hoon Hong

1st-order, 1 variable Problem:

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

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• 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

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• 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

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• 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

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• 2nd-order, 1 variable: Real eigenvalues Problem:

y''C3 y'C2 y = 0 y(0) =K3 y'(0) = 2

1. Find the general solution: λ

2 C3 λC2 = 0

λ =K1, K2

y = C1 e KtCC2 e

K2 t

2. Find the particular solution: y' =KC1 e

KtK2 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

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• Problem: y''K3 y'C2 y = 0 y(0) = 3 y'(0) = 1

1. Find the general solution: λ

2 K3 λC2 = 0

λ = 2, 1

y = C1 e 2 tCC2 e

t

2. Find the particular solution: y' = 2 C1 e

2 tCC2 e t

C1CC2 = 3

2 C1CC2 = 1

C1 =K2, C2 = 5

y =K2 e2 tC5 et

3. Sketch the particular solution:

t

y

t

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• Problem: y''Cy'K6 y = 0 y(0) =K3 y'(0) = 4

1. Find the general solution: λ

2 CλK6 = 0

λ = 2, K3

y = C1 e 2 tCC2 e

K3 t

2. Find the particular solution: y' = 2 C1 e

2 tK3 C2 e K3 t

C1CC2 =K3

2 C1K3 C2 = 4

C1 =K1, C2 =K2

y =Ke2 tK2 eK3 t

3. Sketch the particular solution:

t

y

t

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• Problem: y''C5 y'C6 y = 0 y(0) = 3 y'(0) =K8

1. Find the general solution: λ

2 C5 λC6 = 0

λ =K2, K3

y = C1 e K2 tCC2 e

K3 t

2. Find the particular solution: y' =K2 C1 e

K2 tK3 C2 e K3 t

C1CC2 = 3

K2 C1K3 C2 =K8

C1 = 1, C2 = 2

y = eK2 tC2 eK3 t

3. Sketch the particular solution:

t

y

t

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• Problem: y''C7 y'C12 y = 0 y(0) = 6 y'(0) =K22

1. Find the general solution: λ

2 C7 λC12 = 0

λ =K3, K4

y = C1 e K3 tCC2 e

K4 t

2. Find the particular solution: y' =K3 C1 e

K3 tK4 C2 e K4 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

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• Problem: y''K5 y'C6 y = 0 y(0) = 2 y'(0) = 1

1. Find the general solution: λ

2 K5 λC6 = 0

λ = 3, 2

y = C1 e 3 tCC2 e

2 t

2. Find the particular solution: y' = 3 C1 e

3 tC2 C2 e 2 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: λ

2 CλK2 = 0

λ = 1, K2

y = C1 e tCC2 e

K2 t

2. Find the particular solution: y' = C1 e

tK2 C2 e K2 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: λ

2 C5 λC6 = 0

λ =K2, K3

y = C1 e K2 tCC2 e

K3 t

2. Find the particular solution: y' =K2 C1 e

K2 tK3 C2 e K3 t

C1CC2 =K1

K2 C1K3 C2 = 1

C1 =K2, C2 = 1

y =K2 eK2 tCeK3 t

3. Sketch the particular solution:

t

y

t

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• 2nd-order, 1 variable: Non-real eigenvalues Problem:

y''K4 y'C68 y = 0 y(0) = 1 y'(0) = 10

1. Find the general solution: λ

2 K4 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 Ce

2 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

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• Problem: y''C2 y'C50 y = 0 y(0) = 2 y'(0) = 19

1. Find the general solution: λ

2 C2 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 Ce

Kt 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: λ

2 C64 = 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

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• Problem: y''C2 y'C82 y = 0 y(0) = 0 y'(0) = 9

1. Find the general solution: λ

2 C2 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 Ce

Kt 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

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• Problem: y''K4 y'C125 y = 0 y(0) = 1 y'(0) = 35

1. Find the general solution: λ

2 K4 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 Ce

2 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: λ

2 C9 = 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

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• 1st-order, 2 variables: Introduction Problem:

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

λ 2 KλK6 = 0

λ = 3

v1 = 2 C1 v2 = 4 C1

λ =K2

v1 = 2 C2 v2 =KC2

y1 = 2 C1 e 3 tC2 C2 e

K2 t

y2 = 4 C1 e 3 tKC2 e

K2 t

2. Find the particular solution: 2 C1C2 C2 = 3

4 C1KC2 = 1

C1 = 1 2

, C2 = 1

y1 = e3 tC2 eK2 t

y2 = 2 e3 tKeK2 t

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• 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

λ 2 CλK6 = 0

λ =K3

v1 =K2 C1 v2 =K4 C1

λ = 2

v1 =K2 C2 v2 = C2

y1 =K2 C1 e K3 tK2 C2 e

2 t

y2 =K4 C1 e K3 tCC2 e

2 t

2. Find the particular solution: K2 C1K2 C2 = 3

K4 C1CC2 = 1

C1 =K 1 2

, C2 =K1

y1 = eK3 tC2 e2 t

y2 = 2 eK3 tKe2 t

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• 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

λ 2 C5 λC4 = 0

λ =K1

v1 =K2 C1 v2 = 4 C1

λ =K4

v1 =K2 C2 v2 = C2

y1 =K2 C1 e KtK2 C2 e

K4 t

y2 = 4 C1 e KtCC2 e

K4 t

2. Find the particular solution: K2 C1K2 C2 = 1

4 C1CC2 = 1

C1 = 1 2

, C2 =K1

y1 =KeKtC2 eK4 t

y2 = 2 eKtKeK4 t

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• 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

λ 2 K7 λC10 = 0

λ = 2

v1 =KC1 v2 =KC1

λ = 5

v1 =KC2 v2 = 2 C2

y1 =KC1 e 2 tKC2 e

5 t

y2 =KC1 e 2 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

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• 1st-order, 2 variables: Rea